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E  L  E  M  E  N  U'  S 


EOMETRY   AND   TRIGONOMETRY, 


FROM  THE   WORKS   OI 


A.  M.   LEGENDRE. 


'ISED  AND  ADAPTED  TO  THE  COURSE  OF  MATHEMATICAL  INSTRUCTION  IN 
THE  UNITED  STATES, 


BY   CHARLES    DAYIES,   LL.  D., 

rUOU   OF    ARITHMETIC,    ALGEBRA,    PRACTICAL    MATHEMATICS    FOR    PRACTICAL    MEN, 
ELEMENTS    OP    DESCRIPTIVE    AND    OF     ANALYTICAL    GEOMETRY,    ELEMENTS 
OF    DIFFERENTIAL    AND    INTEGRAL    CALCULUS,    AND    SHADES, 
SILAJDOWS,    AND    PERSPECnVE, 


NEW-YORK: 
PUBLISHED    BY    A.   S.   BARNES    cfe    CO., 


No.   51    JOHN-STREET. 
H.    W 

1856. 


g    C\   r^iyCINJf^I:    H.    W.    DERBY   &   CO. 


D  A  V  I  E  S ' 

COUKSE  OF  MATHEMATICS 


Dabics*  ^[rittnutical  Caiile-JSoofe. 

Babies*  JFirst  Slrssons  in  Slriltjmctic— For  Beginners. 

Dabics'  ^uitl^mftic — Designed  for  the  use  of  Academies  and  Schoola 

Sen  to  3Dabics'  Slntljmctic. 

Babirs'  Snibcrsitn  Slvittimctic— Embracing  the  Science  of  Numbers  and  their 
numerous  Applications. 

Bcw  to  Dabirs'  Bnibcrsitj  S^vitftmctic 

Babies'  Hlcmrntarp  Stlgcbra — Being  an  introduction  to  the  Science,  and  form- 
ing a  connecting  hnk  between  Arithmetic  and  Algebra- 

Sen  to  3Dabirs    HltmcntarP  S^lgrftra. 

Dabffs'  Hlcmrnts  of  Gfomrtrn  axd  ^Tvigononictrn,  with  Applications  in 
Mexscratiox. — Tliis  work  embraces  the  elementary  principles  of  Geometry  and 
Trigonometry.  The  reasoning  is  plain  and  concise,  but  at  the  same  time  strictly 
rigorous. 

Babies'  ^^Dvactiral  fHatljcmatics  for  ^Dvactical  ^rn— Embracing  the  Princi- 
ples of  Drawing,  Architecture,  Meusm-ation,  and  Logarithms,  with  Apphcatioua 
to  the  Mechanic  Arts. 

Babies'  33ouvtJon*S  Algebra — Including  Sturm's  Theorem — Being  an  abridg- 
ment of  the  Work  of  M.  Bourdox,  "with  the  addition  of  practical  examples. 

Babies'  Segrntirc's  G^eometro  axd  2:vigonometrn — From  the  works  of  A.  M. 
Legendre,  with  the  addition  of  a  Treatise  on  Mexsuratiox  of  Plaxes  axd 
Solids,  atid  a  Table  of  Logarithms  and  Logarithmic  Sixes. 

Babies'  Suvbeijing — With  a  description  and  plates  of  the  Theodolite,  Com- 
pass, Plaxe-Table,  and  Level;  also.  Maps  of  the  Topographical  Sigxs  adopted 
by  the  Engineer  Department — an  explanation  of  the  method  of  surveying  the 
Public  Lands,  Geodesic  and  Maritime  Surveying,  and  an  Elementary  Treatise 
on  Navigation. 

Babies'  Bescriptibc  Genmetvn — With  its  application  to  Sphekical  Pbojeo- 

TIuNS. 

Babies'  SljaiJcs,  Sljabotos,  axd  JLinear  ^erspectibc. 

Babies'  Sinalntieal  CJeometrij — Embracing  the  Equatioxs  of  the  Poixt  axd 
Straigiiv  Line — of  tlie  Conic  Sections — of  the  Line  and  Plane  in  Space  ; 
also,  the  discussion  of  the  General  Equatiox  of  the  second  degree,  and  of  Sub- 
faces  of  the  second  order. 

Babies'  Biffercntial  and  i^ntraral  Calculus. 

Babies'  aofiic  aiib  Otiliti)  of  i«att)ematics. 

Entut.f.d  .iccordin<r  to  Act  of  Concrress,  in  the  year  one  thousand  ei^ht  hundred  and 
Unite*. 


liftv-tliree  byCnARLEs  Davies,  in  the  Clerk's  Ollce  of  the  District  Court  of  the 

I'ited  i^tates  for  the  Southern  District  of  New  York. 


J    p.  JOWKS  k  CO,  Sii^BorvK^, 


a?                    PREFACE 
s^  


In  the  preparation  of  the  present  edition  of  the  Geom- 
etry of  A.  M.  Legexdre,  tlie  original  has  been  consulted 
as  a  model  and  guide,  but  not  implicitly  followed  as  a 
standard.  The  language  employed,  and  the  arrangement 
of  the  arguments  in  many  of  the  demonstrations,  will  be 
found  to  differ  essentially  from  the  original,  and  also  from 
the  English  translation  by  Dh.  Brewster. 

.  In  the  original  work,  as  Avell  as  in  the   translation,  the 

^  propositions  are  not  enunciated  in   general  terms,  but  with 
:   reference    to,    and   by  the    aid  of,    the   particular   diagrams 
'    used    for    the    demonstrations.      It    \s    believed    that    this 
^    depai'ture    from    the   method  of  Euclid  has  been  generally 
[    regretted.      The     propositions    of    Gecmietry    are     general 
>^  truths,   and    as    such,  should    be    stated    in    general    terms, 
and  without  reference   to    particular  figures.      The   method 
»    of    enunciating    them   by   the   aid    of    particular   diagrams 
^  seems  to  have  been  adopted   to   avoid  the   difficulty  which 
3   beginners    experience   in    comprehending    abstract   proposi- 
tions.     But  in   avoiding  this  difficulty,  and  thus  lessening, 
at   first,   the   intellectual    labor,  the    faculty   of    abstraction, 
which    it    is    one   of   the   primary  objects  of  the   study  of 
Geometry  to    strengthen,  remains,  to    a    certain  extent,  un 
improved. 


iv  PEEFACE. 

The  methods  of  demonstration,  in  several  of  the  Books, 
have  been  entirely  changed.  By  regarding  the  Circle  as 
the  limit  of  the  inscribed  and  circumscribed  polygons,  the 
demonstrations  in  Book  V.  have  been  much  simplified; 
and  the  same  principle  is  made  the  basis  of  several  im- 
portant demonstrations  in  Book  VIII. 

The  subjects  of  Plane  and  Spherical  Trigonometry 
have  been  treated  in  a  manner  quite  different  from  thai 
employed  in  the  original  work.  In  Plane  Trigonometry, 
especially,  important  changes  have  been  made.  The  sepa- 
ration of  the  part  which  relates  to  the  computations  of  the 
sides  and  angles  of  triangles  from  that  which  is  purely 
analytical,  will,  it  is  hoped,  be  found  to  be  a  decided  im- 
provement. 

The  application  of  Trigonometry  to  the  measurement 
of  Heights  and  Distances,  embracing  the  use  of  the  Table 
of  Logarithms,  and  of  Logarithmic  Sines;  and  the  appli- 
cation of  Geometry  to  the  mensuration  of  planes  and 
solids,  are  useful  exercises  for  the  Student.  Practical  ex- 
amples cannot  fail  to  point  out  the  generality  and  utility 
of   abstract  science. 

Fish  KILL  Landing,  ) 
July,  1851.         I 


CONTENTS. 


PAGB. 

Introduction, ► 0 

BOOK    I. 

Definitions, --  —     13 

Propositions, 21 

BOOK    II. 
Ratios  and  Proportions, 47 

BOOK    III. 

The  Circle,  and  the  Measurement  of  Angles, .     57 

Problems  relating  to  the  First  and  Third  Books, 76 

BOOK    IV. 

Proportions  of  Figures — Measurement  of  Areas, 87 

Problems  relating  to  the  Fourth  Book, 122 

BOOK    V. 
Regular  Polygons — Measurement  of  the  Circle, 135 

BOOK    VI. 
Planes  and  Polyedral  Angles 156 

BOOK    VII. 
Polyedrons, ^ 174 

BOOK    VIII. 
The  Three  Round  Bodies, 203 

BOOK    IX. 
Spherical  Geometry, ». ....- 227 


vi  CONTENTS, 


APPENDIX. 


PAGE. 


Note  A, 21 

The  Regulai  Polyedrons, 217 

Application  of  Algebra  to  the  Soluiion  of  Lxeo'neincai  Problems, 219 


PLANE    TEIGONO:\[ETPvY. 

Logarithms  Defined, 255 

Logarithms,  Use  of, 25G 

General  Principles, 256 

Table  of  Logarithms, 257 

To  Find  from  the  Table  the  Logarithm  of  a  Number, 258 

To  Find  from  the  Table  the  Number  corresponding  to  a  Given  Loga- 
rithm   260 

Multiplication  by  Logarithms, 261 

Division  by  Logarithms, 262 

Arithmetical  Complement, 263 

To  find  the  Powers  and  Roots  of  Numbera^  by  Logarithms, 265 

Geometrical  Constructions, 2G6 

Description  of  Instruments, ' 266 

Dividers, 266 

Ruler  and  Triangle, 260 

Problems, 267 

Scale  of  Equal  Parts, 268 

Diagonal  Scale  of  Equal  Parts, 268 

Semicircular  Protractor, 270 

To  Lay  off  an  Angle  with  a  Protrax;tor, 270 

c'arts  of  a  Plane  Triangle, 271 

Plane  Trigonometry,  Defined 271 

Division  of  the  Circumference, 271 

Measures  of  Angles, 271 

Complement  of  an  Arc, 271 

Definitions  of  Trigonometrical   Lines, 272 

Table  of  Natural  Sines, 273 

Table  of  Logarithmic  Sines, 274 

To  Find  from  the  Table, the  Logarithmic  Sine,  &c.,  of  an  Arc  or  Angle,  274 
To  Find  the  Degrees,  &c..  Answering  to  a  Given  Logarithmic  Sine,  &c.,  276 

Theorems, !. 277 

Solution  of  Triangles, 281 

Solution  of  Right-Angled  Triangles, 287 

Application  to  Heights  and  Distances, 288 


CONTENTS.  vii 


ANALYTICAL    PLANE    TEIGONOMETRY. 

PAGE. 

Circular  Functions, 297 

Analytical  Plane  Trigonometry,  Defined, 297 

Quadrants  of  the  Circumference, 298 

Versed-Sine, 298 

Relations  of  Circular  Functions, 299 

Table  I.,  of  Formulas,. 301 

Algebraic  Signs  of  the  Functions, 301 

Table  II.,  of  Formulas, 306 

General  Formulas, 307 

Homogeneity  of  Terms, 313 

Formulas  for  Triangles, 315 

Construction  of  Trigonometrical  Tables, 317 


SPHEEICAL    TEIGONOMETRY. 

Spherical  Triangle,  Defined,. 321 

Spherical  Trigonometry,  Defined, 321 

First  Principles, 32 J 

Napier's  Analogies, 329 

Napier's  Circular  Parts, 329 

Theorems, 330 

Solution  of  Right-Angled  Spherical  Triangles,  by  Logarithni.s, 333 

Of  Quadrantal  Triangles, ^2-^ 

Solution  of  Oblique-Angled  Triangles,  by  Logarithms,. .   ..    S?^ 

MENSURATION    OF    SUEFAOES. 

Area,  or  Contents  of  a  Surface, 34? 

Unit  of  Measure  for  Surfaces, 34"^ 

Area  of  a  Square,  Rectangle,  or  Parallelogram, 34''' 

Ar«a  of  a  Triangle, 348 

Area  of  a  Trapezoid, 350 

Area  of  a  Quadrilateral, 35^ 

Area  of  an  Irregular  Polygon, 35 J 

Area  of  a  Long  and  Irregular  Figure  bounded  Oii  One  Side  by  a  Right 

Line, 35 

Area  of  a  Regular  Polygon, 353 

To  Find  the  Circumference  or  Diameter  of  a  Circle, 354 

T .  find  the  Length  of  an  Arc, 355 

Area  of  a  Circle, 356 

Area  of  a  Sector  of  a  Circle, 356 

Area  of  a  Segment  of  a  Circle, 356 

Area  of  l  Circular  Ring, 357 


viii  CONTENTS. 

» 

MENSUEATION    OF    SOLIDS. 

PAGE. 

Mensuration  of  Solids,  divided  into  Two  Parts, 358 

Unit  of  Length, 358 

Unit  of  Solidity, 358 

Table  of  Solid  Measures, 358 

Surface  of  a  Right  Prism, 358 

Surface  of  a  Right  Pyramid, 359 

Convex  Surface  of  the  Frustum  of  a  Right  Pyramid, 359 

Solidity  of  a  Prism, 359 

Solidity  of  a  Pyramid, 360 

Solidity  of  the  Frustum  of  a  Pyramid, 360 

The  Wedge, 361 

Rectangular  Prismoid, 361 

Solidity  of  the  Wedge, 361 

Solidity  of  a  Rectangular  Prismoid, 362 

Surface  of  a  Cylinder, 363 

Convex  Surface  of  a  Cone, 364 

Surface  of  a  Frustum  of  a  Cone, 364 

Solidity  of  a  Cylinder, 364 

Solidity  of  a  Cone, 365 

Solidity  of  a  Frustum  of  a  Cone, 365 

Surface  of  a  Spherical  Zone, 365 

Solidity  of  a  Sphere, 366 

Solidity  of  a  Spherical  Segment, 366 

Surface  of  a  Spherical  Triangle, 366 

Surface  of  a  Spherical  Polygon, 367 

Of  the  Regular  Polyedrons, 367 

Theorem, 367 

Method  of  Finding  the  Angle  included  between  two  Adjacent  Faces  of 

a  Regular  Polyedron, 368 

Table  of  R egular  Polyedrons  whose  Edges  are  1, 369 

Solidity  of  a  Ri;gular  Polyedron, 309 


ELEMENTS 


OF 


GEOMETRY. 


I'N'TRODUCTION. 


1.  Space  extends  indefinitely  in  every  direction  and 
contains  all  bodies. 

2.  Extension  is  a  limited  portion  of  space,  and  hiis 
three  dimensions,  length,  breadth,  and  thickness. 

3.  A  Solid,  or  Body,  is  a  liioited  portion  of  space 
supposed  to  be  occupied  by  matter.  The  difference  be- 
tween the  terms,  extension  and  solid^  is  simply  this  :  the 
former  denotes  a  limited  portion  of  space,  viewed  in  the 
abstract,  while  the  latter  denotes  such  a  portion  occupied 
by  matter. 

The  term,  Solid^  is  generally  used  in  Geometry,  in  pre- 
ference to  Extension,  because  the  mind  apprehends  readily 
the  forms  and  relations  of  tangible  objects,  while  it  often 
experiences  much  difficulty  in  dealing  with  the  abstract 
notions  derived  from  them.  It  is,  however,  important  to 
observe,  that  the  geometrical  properties  of  solids  have  no  con- 
nection whatever  ivith  matter^  and  that  the  demonstrations  which 
establish  and  make  known  those  properties^  are  based  on  Hce 
attributes  of  extension  only. 


10  GEOMETEY. 

4.  A  Solid  being  a  limited  portion  of  space,  is  nov^cjs- 
sarily  divided  from  the  indefinite  s])ace  which  siirrounclM 
it :  that  which  so  divides  it,  is  called  a  Surface.  Now, 
since  that  which  bounds  a  solid  is  no  part  of  the  solid 
itself,    it   follows,   that   a   surface   has   but   two   dimensions, 

ongth  and  breadth. 

5.  If  we  consider  a  limited  portion  of  a  su]face,  that 
which  separates  sach  portion  from  the  other  parts  of 
the  surface,  is  called  a  Line.  This  mark  of  division 
forms  no  part  of  the  surfaces  which  it  separates  :  hence,  a 
line  has  length  only,  without  breadth  or  thickness. 

6.  If  we  regard  a  limited  portion  of  a  line,  that  which 
5epariites  such  portion  from  the  part,  at  either  extremity, 
is  called  a  Point.  But  this  mark  of  division  forms  no 
part  of  the  line  itself:  hence,  a  point  has  neither  length, 
breadth,  nor  thickness,  but  place  or  position  only. 

7.  Although  we  use  the  term  solid  to  denote  a  given 
portion  of  space,  the  term  surface  to  denote  the  boundary 
of  a  solid,  the  term  line  to  denote  the  boundary  of  a  sur- 
face, and  the  term  ]^oint  to  designate  the  limit  of  a  line, 
still,  we  may  employ  either  of  these  terms,  in  an  abstract 
sense,  without  any  reference  to  the  others. 

Thus,  we  may  contemplate  a  river,  as  ^  solid,  without 
considering  its  boundaries ;  may  look  upon  the  surface  and 
perceive  that  it  has  length  ajid  breadth  without  refering  to 
its  depth ;  or,  we  may  regard  the  distance  across  without 
taking  into  account  either  its  depth  or  length.  So  like- 
wise, we  may  consider  a  point  vfithout  any  reference  to 
the  line  which  it  limits. 

In  the  definitions  and  reasonings  of  Geometry  these 
terms  are  always  used  in  an  abstract  sense;  they  are  mere 
signs  to  the  mind  of  the  conceptions  for  which  they  stand. 

8.  AxGLE  is  a  term  which  designates  the  portion  of  a 
surface  included  by  two  lines  meeting  at  a  common  point  ; 


INTRODUCTION.  11 

and  it  also  denotes  a  portion  of  space  included  by  two  or 
more  planes. 

9.  Magnitude  is  a  general  term  employed  to  denote 
tliose   quantities   whicli   arise   from   considering  the  dimen- 

ions    of    extension,    and    is    equally    applicable    to    Lines, 
ii;2;les,   surfaces,  and  solids.      Geometry  is  conversant   "vvith 
four  kinds  of  magnitude. 

1.  Lines;  which  have  length  without  breadth  or  thickness. 

2.  Angles ;  bounded  by  straight  lines,  by  curves,  and  by 
planes. 

3.  Surfaces  ;  which  have  length  and  breadth  without 
thickness :   and 

4.  Solids ;    which  have  length,  breadth,  and  thickness. 

10.  Figure  is  a  term  applied  to  a  geometrical  magni- 
tude and  expresses  the  idea  of  shape  or  form.  It  is  that 
which  is  enclosed  by  one  or  more  boundaries.  Thus,  "  A 
triangle  is  a  plane  figure  bounded  by  three  straight  lines." 

11.  A  Property  of  a  figure  is  a  mark  or  attribute 
common  to  all  figures  of  the  same  class. 

12.  The  portions  of  extension  which  constitute  the  geo- 
metrical magnitudes,  are  indicated  to  the  mind  by  cer- 
tain  marks   called  lines. 

Thus,  we  say,  the  straight  line  AB^  is  the  shortest  dis- 
tance between  the  two  points  A  and 

B.      The  mark  AB^  on  the  paper,  is  A B 

not  the  geometrical  line  AB,  but  only 

the  sign  or  representative  of  it — the  geometrical  line  itseli^ 

having   merely  a  mental  existence. 

We  also  say,  that  the  triangle 
ACB  is  bounded  by  the  three  straight 
lines  AB^  A  (7,  CB,  Now,  the  triangle 
ACB^  is  but  the  sign,  to  the  mind, 
of  a  portion  of  a  plane.     That  which 

the  eye   sees   is  not  the   geometrical  j^ g 

conception   on   which   the  mind  acts 

and   reasons  :     but   is,  as  it  were,  the  w<»rd  or  sign  which 

stands  for   and   expresses  the  abstract  idea. 


12  GEOMETEY. 

These  considerations  have  induced  me  to  represent  tlie 
geometrical  magnitudes  by  the  fewest  possible  lines,  and  to 
reject  altogether  the  method  of  shading  the  figures.  It  is 
the  conception  of  extension,  in  the  abstract,  with  which 
the  mind  should  be  made  conversant,  and  too  much  pains 
cannot  be  taken  to  exclude  the  idea  that  we  are  dealing 
with  material  tluncfs!. 


ELEMENTS  OF  GEOMETRY 


BOOK  I. 

DEFINITIONS.* 


1.  Extension    lias    three    dimensions,   length,   breadth, 
and  thickness. 

2.  Geometry  is  the  science  which  has  for  its  object : 
1st.   The   measurement  of '  extension  ;    and  2dlj.  To  dia- 

cover,  by  means  of  such  measurement,  the   properties  and 
relations  of  geometrical  magnitudes. 

8.   A  Point  is  that  which  has  place,  or  position,  buk 
not  magnitude. 

4.   A  Line  is  length,  without  breadth  or  thickness. 

6.   A    Straight   Line  is  one  which 

lies  in  the  same  direction  between  any 

two  of  its  points. 


6.  A  Broken  Line  is  one  made  up 
of  straight  lines,  not  lying  in  the  same 
direction. 


7.  A  Curve    Line    is    one    which 
changes  its  direction  at  every  point. 

The  word  line  when  used  alone,  will  designate  a  straight 
line ;    and  the  word  curve^  a  curve  line. 

8.  A  Surface   is   that   which  has  length  and  breadth 
without  thickness. 


*  Soo  Davicfi'  Logig  and  Utility  of  Matliein.ition.  ^  1. 


14 


GEOMETRY. 


9.  A  Plaxe  is  a- surface,  such,  that  if  any  two  of  its 
points  be  joined  by  a  straight  line,  such  line  will  be  wholly 
hi  the  surface. 

10.  Every  surface,  which  is  not  a  plane  sui^face,  or  con: 
posed  of  plane  surfaces,  is  a  curved  surface. 

11.  A  Solid,  or  Body  is  that  which  has  length,  breadth, 
and  thickness :  it  therefore  combines  the  three  dimensions 
of  extension. 

12.  A  plane  Angle  is  the  portion  of  a  plane  included 
between  two  straight  lines  meeting  at  a  common  point. 
The  two  straight  Hues  are  called  the  sides  of  the  angle, 
and  the  common  point  of  intersection,  the  vertex. 

Thus,  the  part  of  the  j^lane  includ- 
ed between  AB  and  AC  is  called  an 
angle :  AB  and  A  C  are  its  sides,  and  A 
its  vertex. 

An   angle   is    sometimes    designated 
simply  by  a  letter  placed  at  the  vertex, 
as,  the   angle  A  ;    but  generally,   by   three   letters,    as,    tne 
angle  BAC  or  CAB, — the  letter  at  the  vertex  being  always 
placed  in  the  middle. 


13.  When  a  straight  line  meets  an- 
other straight  line,  so  as  to  make  the 
Adjacent  angles  equal  to  each  other, 
each  angle  is  called  a  rir/ht  angle  ;  and 
the  first  line  is  said  to  be  perpendtcu- 
lar  to  the  second. 


14.   An  Acute  Angle  is  an  angle 
xsa  than  a  right  angle. 


15.   An  Obtuse  Angle  is  an  angle 
greater  than  a  right  angle. 


~^' 


X 


BOOK    I.  16 

16.  Two  straight  lines  are  said  to 

be   parallel^     when   being    situated   in 

the  same  plane,  they  cannot  meet,  how 
far  soever,  either  way,  both  of  them 
be  produced. 

17.  A  Plake  Figure  is  a  portion  of  a  plane  terminat- 
ed on  all  sides  by  lines,  either  straight  or  curved. 

18.  A  Polygon,  or  rectilineal  fig- 
ure^ is  a  portion  of  a  plane  terminat- 
ed on  all  sides  by  straight  lines. 

The   broken   line   that   bounds    a 
polygon  is  called  its  perimeter. 


19.  The  polygon  of  three  sides,  the  simplest  of  all,  Ls 
called  a  triangle;  that  of  four  sides,  a  quadrilateral;  that 
of  five,  a  ^pentagon ;  that  of  six,  a  hexagon ;  that  of  seven. 
a  heptagon ;  that  of  eight,  an  octagon ;  that  of  nine,  an 
nonagon  ;  that  of  ten,  a  decagon ;  and  that  of  twelve,  a 
iodecogon. 

20.  An  Equilateral  polygon  is  one  which  has  all  its 
sides  equal ;  an  equiangular  polygon,  is  one  which  has  all 
its  angles  equal. 

21.  Two  polygons  are  equilateral.,  or  mutually  equilateral 
when  they  have  their  sides  equal  each  to  each,  and  placed 
hi  the  same  order :  that  is  to  say,  when  following  their 
bounding  lines  in  the  same  direction,  the  first  side  of  the 
one  is  equal  to  the  first  side  of  the  other,  the  second  to 
the  second,  the  third  to  the  third,  and  so  on. 

22.  Two  polygons  are  equiangular.,  or  mutually  equiangu- 
lar., when  every  angle  of  the  one  is  equal  to  a  correspond- 
ing angle  of  the  other,  each  to  each. 

23.  Triangles  are  divided  into  classes  with  reference 
both  to  their  sides  and  angles. 

1.    An    equilateral    triangle    is    one 
which  has  its  three  sides  equal. 


10 


GEOMETRY 


2.  An    isosceles  triangle   is  one   whicli 
has  two  of  its  sides  equal. 


8.   A  scalene  triangle  is  one  whicli  has 
\i&  three  sides  unequal. 


4.    An    acute-angled    triangle    is     one 
which  has  its  three  angles  acute. 


5.  A  right-angled  triangle  is  one  which 
has  a  right  angle.  The  side  opposite  the 
right  angle  is  called  the  hypothenuse,  and 
the  other  two  sides,  the  base  and  perpen- 
dicula: 


6.    An     obtuse-angled     triangle    is    one 
vvhich  has  an  obtuse  angle. 


2-4.   There  are  three  kinds  of  Quadrilaterals 


^        1.    Tlie  trapezium^  which  has  no  two 
of  its  sides  parallel. 


2.    Tlie  trapezoid^  which  has  only  two 
of  its  sides  parallel. 


8.    The  parallelogram^   which  has   its 
opposite  sides  parallel. 


BOOK  I.  17 

25.   There  are  four  varieties  of  Parallelograms: 


1.    The  rho/nhoid,  wliicli  lias  no  riglit 
angle. 


2.    The   rhomhuSj  or  loze^ige,  wliicli   is 
an  equilateral  rhomboid. 


3.    The  rectangle,  which  is  an  eqiiian- 


7 


gular  parallelogram. 


"^    4.    The  square,  which  is  both  equilat- 
eral and  equiangular. 


26.  A   Diagonal  of  a  figure  is  a  line  which  joins  the 
vertices  of  two  angles  not  adjacent. 

27.  A  base  of  a  plane  figure  is  one  of  its  sides  on  whicl) 
it  may  be  supposed  to  stand. 


DEFINITIONS   OF   TERMS. 

1.  An  axiom  is  a  self-evident  truth. 

2.  A  demonstration  is  a  train  of  logical  argnments  brought 
to  a  conclusion. 

3.  A  theorem  is  a  truth  which  becomes  evident  by  means 
of  a  demonstration. 

4.  A  problem  is  a  question   proposed,  which  requires  a 
solution. 

5.  A   lemma  is   a  subsidiary  truth,    employed    for   the 
demonstration  of  a  theorem,  or  the   solution  of  a  probleiiu 

2 


18  GEOMETRY. 

6.  The  common  name,  proposition^  is  applied  mdifferent- 
ly,  to  theorems,  problems,  and  lemmas. 

7.  A  corollary  is  an  obvious  consequence,  deduced  from 
one  or  several  propositions. 

8.  A  scholium  is  a  remark  on  one  or  several  preceding 
ropositions,    which   tends   to    point    out    their    connection, 

their  use,  their  restriction,  or  their  extension. 

9.  A  hypothesis  is  a  supposition,  made  either  in  the 
enunciation  of  a  proposition,  or  in  the  course  of  a  demon- 
stration. 

10.  A  postulate  grants  the  solution  of  a  self-evident 
problem. 

'"-*       EXPLAXATIOX   OF   SIGXS 

1.  The  sign  =  is  the  sign  of  equality ;  thus,  the  ex- 
pression A  =  B^  signifies  that  A  is  equal  to  B. 

2.  To  signify  that  A  is  smaller  than  B^  the  expression 
A  <  ^  is  used, 

8.  To  signify  that  A  is  greater  than  B^  the  expression 
J.  >  ^  is  used ;  the  smaller  quantity  being  always  at  the 
vertex  of  the  angle. 

4.  The  sign  +  is  called  plus  ;  it  indicates  addition . 

5.  The  sign  —  is  called  minus  ;  it  indicates  subtraction : 
Thus,    A-\-B^    represents   the   sum   of  the    quantities  A 

and  j5;  A  —  B  represents  their  difference,  or  what  remains 
after  B  is  taken  from  A\  and  A—B-^C,  ot  A+C—B,  sig- 
nifies that  A  and  C  are  to  be  added  together,  and  that  B 
is  to  be  subtracted  from  their  sum. 

-^    6.    The   sign  X  indicates  multiplication:   thus  J. Xj5  re- 
presents the  product  of  A  and  B. 

The  expression  Ax{B-\-C—D)  represents  the  product  of 
A  by  the  quantity  B  +  C-D.  l^  A+D  were  to  be  multi- 
plied by  A—B-\-C^  the   product  woukl  be  indicated   thus; 

{A+D)X{A-B  +  C\ 

whatever  is  enclosed  within  the  curved  lines,  being  consid- 


BOOK   I.  19 

ered  as  a  single   qaantity.     The  same   thing  may   also  be 
indicated  by  a  bar:   thus, 


A+B+GxD, 

denotes  that  the  sum   of  A^  B  and  C,  is  to  be  multiplied 
by  D. 

7.  A  figure  placed  before  a  line,  or  quantity,  serves 
as  a  multiplier  to  that  line  or  quantity ;  thus  ^AB  signifies 
that  the  line  AB  is  taken  three  times ;  \A  signifies  the 
half  of  the  angle  A. 

8.  The  square  of  the  line  AB  is  designated  by  AB"] 
its  cube  by  AB' .  What  is  meant  by  the  square  and  cube 
of  ii  line,  will  be  explained  in  its  proper  place. 

9.  The  sign  \/  indicates  a  root  to  be  extracted ;  thus  \/2 
means  the  square-root  of  2 ;  \/AxB  means  the  square-root 
of  the  ]Droduct  of  A  and  B. 

AXIOMS. 

1.  Things  which  are  equal  to  the  same  thing,  are  equal 
to  one  another. 

2.  If  equals  be  added  to  equals,  the  wholes  will  be 
equal. 

8.  If  equals  be  taken  from  equals,  the  remainders  ^noII 
be  equal. 

4.  If  equals  be  added  to  unequals,  the  wholes  will  be 
unequal. 

5.  If  equals  be  taken  from  unequals,  the  remainders 
will  be  unequal. 

6.  Things  which  are  doubles  of  equal  things,  are  equal 
to  each  other. 

7.  Things  which  are  halves  of  equal  things,  are  equaJ. 
to  each  other. 

8.  The  whole  is  greater  than  any  of  its  parts. 

9.  The  whole  is  equal  to  the  sum  of  all  its  parts. 


^ 


20  GEOMETRY. 

10  All  right  angles  are  equal  to  each,  otlier. 

11  l^'rom   one  point  to   anotlier   only  one  straight  line 
can  be  drawn. 

12.  A  straight  line  is  the  shortest  distance  between  tvvo 
points. 

13.  Through  the  same  point,  only  one  straight  line  can 
be  drawn  which  shall  be  parallel  to  a  given  line. 

11:.    Magnitudes,  which   being    applied   the    one    to   the 
other,  coincide  throughout  their  whole  extent,  are  equal 

^  POSTULATES. 

1.  Let  it  be  granted,  that  a  straight  line  may  be  drawn 
from  one  point  to  another  point. 

2.  That  a  terminated    straight   line   may  be   prolonged, 
in  a  straight  line,  to  any  length. 

3.  That  if  two  straight  lines  are  unequal,  the  length  of 
the  less  may  always  be  laid  off  on  the  greater. 

4.  That  a  given  straight  line  may  be 

bisected:   that  is,  divided  into  two  equal 

parts. 


5.    That    a    straight    line   may  bisect 
a  given  angle. 


6.  That  a  perpendicular  may  be 
drawn  to  a  given  straight  hne,  either 
from  a  point  without  the  line,  or  at  a 
point  of  a  line. 

7.  That  a  straight  line-  may  be 
dra'vvn,  making  with  a  given  straight 
line,  an   angle   equal  to   a  given  angle. 


BOOK   I. 


21 


FKOPOSITION   I.     TIIEOREiL 


If  oiie  straight  line  meet   another   straight   line,  the.   sura  of  the 
tivo  adjacent  angles  ivill  he  equal  to  two  right  angles. 

Let  the  straight  line  DC  meet  the  straight  line  AB  at  (7; 
tben  will  the  angle  A  CD  plus  the  angle  DCB^  be  et^ual  to 
two  right  angles.  _ 

At  the  point  C  suppose  CE  to 
be  drawn  perpendicular  to  JiB : 
then,  ACE  4-  ECB  =  two  riglit 
angles  (i).  18).^  But  ECB  is  equal 
to  ECD  -h  DCB  (a.  9)  :  hence, 
ACE    +    ECD    -f    DCB   =    two 

right  angles.  B-at  ACE  +  ECD  =  ACD  (a.  9):  there- 
fore, ACD  ■\-  DCB  =  two  right  angles. 

Cor.  1.     If  one  of  the  angles  ACD   or  DCB^  is  a  righi 
angle,  the  other  will  also  be  a  right  angle. 

Cor,  2.    If  a   straight    line    DE  ^ 

is  perpendicular  to  another  straight 

line  AB\  then,  reciprocally,  AB  will      ^ i ^ 

be  perpendicular  to  DE. 

For,  since  DE  is  perpendicular 
to  AB^  the    angle  ACD  will  be   a  E 

right  angle  (d.  13).  But  since  AC  meets  DE  at  the  point 
(7,  making  one  angle  ACD  a  right  angle,  the  adjacent  angle 
ACE  will  also  be  a  right  angle  (c.  1).  Therefore,  AB  is 
perpendicular   to   DE  (d.  13). 

C(fr.  3.  The  sum  of  the  succes- 
sive angles  BAC,  CAD,  DAE,  EAE, 
formed  on  the  same  side  of  the 
line  BF,  is  equal  to  two  right  an- 
gles ;  for,  tlieir  sum  is  equal  to  ^  A 
that  of  the  two  adjacent  angles  BA  C  and  CAF. 


*Tn  tlie  references,  A.  stands  for  Axiom— D.  f(;r  Definition— B.  for  Book- -P.  int 
Froijositiiui — C  for  Corollary — S.  for  Sclioliuin,  auJ  Prob.  for  Problem. 


22  GEOMETRY, 


PEOPOSITION  II.     TIIEOEEM. 

Two  straight  lines,  which  have  two  j^oints  common,  coincide  the 
one  with  the  othei',  thrmighoiit  their  whole  extent,  and  fimm 
one  and  the  same  straight  line. 

Let  A  and  B  be  the  two  common  points  of  two  straigli 
lines.  .        ^ 

In  the  first  place,  the  two  lines 
will  coincide  between  the  points  A 
and  B ;  for,  otherwise  there  would 
be  two  straight  lines  between  A  and 
Bj  which  is  impossible  (a.  11). 

Suj^pose,  however,  that  in  being  prolonged,  these  lines 
begin  to  separate  at  some  point,  as  (7,  the  one  becoming 
CZ),  the  other,  CE.  At  the  point  C,  suppose  CF  to  be 
drawn,  makinof  with  AC,  the  rioht  an^le  ACF. 

Now,  since  ACD  is  a  straight  line,  the  angle  FCD  will 
be  a  right  angle  (p.  I.,  c.  1) :  and  since  A  CE  is  a  straight 
line,  the  angle  FCE  will  also  be  a  right  angle.  Ilence, 
the  angle  FCD  is  equal  to  the  angle  FCE  (a.  10) :  that  is, 
a  whole  is  equal  to  one  of  its  parts,  which  is  impossible 
(a.  8) :  therefore  the  two  straight  lines  which  have  two 
points,  A  and  i?,  in  common,  cannot  separate  at  any  point, 
when  prolonged ;  hence,  they  form  one  and  the  same 
straight  line."^ 

PKOPOSITION    III.      TIIEOEEM. 

If^  when  a  straight  line  meets  two  other  straight  lines  at  a 
common  jwint^  the  sum  of  tlie  two  adjacent  angles  icJiich  it 
makes  icith  thevi,  is  equal  to  two  rigid  angles,  tJie  two 
straight  lines  which  are  met,  form  one  and  the  same  straiglU 
line. 

Let  the  straight  line  CD  meet  the  two  lines  AC,  CB,  at 
their  o^mmon  point  C,  and  let  the  sum  of  the  tAvo  adja- 
cent angles,  DC  A,  DCB,  be  equal  to  two  right  angles:  then 


*  See  Note  A.    It  is  earnestly  recommended  to  every  pupil  to  read  and  undur- 
stand  this  Note.    Also,  see  Logic  and  Utility  of  Mathematics,  §  262. 


BOOK    T.  23 

will  CB  be  tlie  prolongation  oi  AC]    oi^  AC  and  CB  will 
form  one  and  tlie  same  straight  line. 

For,  if  CB  is  not  the  prolonga- 
tion of  AC^  let  CE  be  that  prolon- 
gation. Then  the  line  ACE  being 
straight,  the  sum  of  the  angles  A  CD, 
BCE,  will  be  equal  to  two  right 
angles  (p.  i).  But  by  hypothesis, 
the  sum  of  the  angles  A  CD,  ECB, 
is  also  equal  to  two  right  angles: 
therefore  (a.  1), 

ACD-{-DCE  must  be  equal  to  ACD+BCB. 
Taking  away  the  angle  A  CD  from  each,  there  remains  the 
angle   BCE   equal   to    the    angle    DCB :   that  is,    a  whole 
equal  to  a  part,  which  is  impossible   (a.  8) :    therefore,  A  C 
and  CB  form  one  and  the  same  straight  line. 

PEOPOSITION    IV.     TIIEOEEM. 

When   two   straight    lines   intersect  each   other,    the   opposite   or 
vertical  angles,  which  they  form,  are  equal. 

Let  AB  and  DE  be  two  straight  lines,  intersecting  each 
other  at  C;  then  will  the  angle  ECB  be  equal  to  the 
angle  ACD,  and  the  angle  ACE  to  the  angle  DCB. 

For,  since  the  straight  line  DE     A> 
is    met    by   the    straight    line   AC, 
the  sum  of  the  angles  ACE,  A  CD, 
is  equal  to  two  right  angles  (p.  i.);     j-). 
and  since   the   straio^ht  line  AB  is 
met  by  the  straight  line  EC,  the  sum  of  the   angles  ACE, 
and  ECB,  is  equal  to  two  right  angles :   hence  (a.  1), 

ACE-hACD  is  equal  to  ACE-hECB. 
Take   away  from    both,    the    common    angle    ACE,    there 
remains    (a.  8)  the   angle   A  CD,    equal   to   its    opposite    or 
vertical  angle  ECB.     In  a  similar  manner  it  may  be  proved 
that  ACE  is  equal  to  DCB. 

Scholium.  The  four  angles  formed  about  a  point  by  two 
Straight  lines,  which  intersect  each  other,  are  together  equal 


24 


^i^OMETRY, 


to  four  right  angles.  For,  the  sum  of  the  two  angles  ACI^^ 
ECB^  is  equal  to  two  right  angles  (p.  i) ;  and  the  sum  of 
the    other   two,    ACD^    1)CJ\    is   also    equal    to    two    right 


angles 


therefore,    the   sum   of  the   four,  is   equal    to   four 


right  angles. 

In  general,  if  any  numler  of  straight 
lines  CA,  CB,  CD,  &;c.,  meet  in  a  com- 
mon point  C,  the  sum  of  all  the  suc- 
cessive angles,  ACB,  BCD,  DCE,  EOF, 
FCA,  will  be  equal  to  four  right  an- 
gles. For,  if  four  right  angles  were 
formed  about  the  point  C,  by  two  lines 
perpendicular  to  each  other,  their  sum  would  be  equal  to  the 
sum  of  the  successive  angles  ACB,  BCD,  DCE,  ECF,  FCA, 


PEOPOSITIOX   V.      TIIEOEEM. 

If  two  triangles  have  two  sides  and  the  included  angle  of  the 
one,  equal  to  tivo  sides  and  the  included  angle  of  the  othei', 
each  to  each,  the  two  triangles  will  he  equal. 

In  the  two  triangles  EDF  and  BAC,  let  the  side  EL 
be  equal  to  the  side  BA^  the  side  DF  to  the  side  A  (7,  and 
the  angle  D  to  the  angle  A  ;  then  will  the  triangle  EDF 
be  equal  to  the  triangle  BAC. 

For,  if  these  trian- 
gles be  applied  the  one 
to  the  other,  they  will 
exactly   coincide.     Let 

the  side  ED  be  placed 

on  the  equal  side  BA;     ^  ^  ^  ^ 

then,  sin;if3  the  angle  D  is  equal  to  the  angle  A^  the  side 
DF  will  take  the  direction  A  C.  But  DF  is  equal  to  A  G\ 
therefore  the  point  F  will  fall  on  (7,  and  the  third  side  EF^ 
will  coincide  with  the  third  side  BC  (a.  11) :  consequent- 
ly, the  triangle  EDF  is  equal  to  the  triangle  BAC  (a.  14). 

Cor.  When  two  triangles  have  these  three  things  equal, 
viz.,  the  side  ED=BA,  the  side  DF=AC,  and  tlie  anglo 
Z)=^4,  the  remaining  three  are  also  respectively  equal,  viz., 
the  side  EF=BC,  the  angle  E=B,  and  the  angle  F-^C. 


BOOK    I.  25 


PROPOSITION    Vr.     THEOREM. 

If  two  trmngles  have  two  aiKjIes  and  the  indmyi  .side  of  iJic 
one,  equal  to  tuso  anjles  and  the  incluiUd  .side  of  Inc  other^ 
each   to  each,  the  two  trijiiKjtes  ivill  he  e'j'ud. 

Let  IWF  and  BAC  be  two  triangles,  having  the  angle 
£J  eqnal  to  the  angle  />,  the  angle  F  to  the  angle  C\  and 
the  included  side  FF  to  the  included  side  />6';  ther  will 
the  triangle  FDF  be  equal  to  the  triani-le  BAG. 

For,  let  the  side  FF 
be  placed  on  its  equal 
BC,  the  i)oint  F  falling 
on  IJ,  and  the  point  F'  on 
C.  Then,  since  the  angle 
E  is  e(pial  to  the  angle  ^ 
B,  the  side  FD  will  take  the  direction  BA  ;  and  hence, 
the  ])oint  J)  will  be  found  somewhere  in  the  line  5.4.  In 
like  manner,  since  the  angle  F  is  equal  to  the  angle  C, 
the  line  FD  will  take  the  direction  CA,  and  the  point  I) 
will  be  found  somewhere  in  the  line  CA.  Hence,  the 
point  I),  falling  at  the  same  time  in  the  two  straight  lines 
BA  and  6V1,  must  fall  at  their  intersection  A  :  hence,  the 
two  triangles  FDF^  BAQ  coincide  with  each  other,  and 
consequently,  are  equal  (a.   14). 

Cor.  Whenever,  in  two  triangles,  these  three  things  are 
equal,  viz.:  the  angle  F=B,  tlie  angle  F=C^  and  the 
included  side  FF  equal  to  the  included  side  BO,  it  may 
be  inferred  that  the  remaining  three  are  also  respectively 
equal,  viz. :  the  angle  1)—A,  the  side  FJJ=BA^  and  the 
side  JJF=AC. 

SclioUum.  Two  triangles  which  being  applied  to  each 
other,  coincide  in  all  their  parts,  are  equal  (a.  14).  The 
like  parts  are  those  which  coincide  with  each  other;  hence, 
they  are  also  equal  each  to  each.  The  converse  of  this 
proposition  is  also  true ;  viz.,  if  two  trianr/Ies  hare  all  the 
'parts  cf  the  one  eqnal  to  the  imrts  of  the  other,  (ach  to  eac\ 
the  triangles  will  he  eqnal:  for,  when  applied  to  each  other, 
they  will  mutually  coincide. 


26  GEOMETEY. 


PROPOSITION  VII.    THEOREM. 

The  sum  of  any   two  sides   of  a   triangle^  is  greater   tfMn  the 

tlurd  side. 

Let  ABC  be  a  triangle  :    tlien   Avill  the   sum  of  two  of 
its  sides,  as  AB^  BC^  be  greater  than  the  third  side  AC 

B 

For  the  straight  line  J.  (7  is  the  short- 
est distance  between  the  points  A  and  C 
(a.  12);  hence,  Ai>H-i>  (7  is  greater  than 
AC.        '  A 

Cor.   If  from  both  members  of  the  inequality 

AC<:AB+BC 

we    take    away   either  of  the   sides,  as  BC^  we  shall   have 
(A.  5) 

AC-BC<AB: 

that  is,  the    difference   hetween    any    tico    sides   of  a  triangle  is 
less  titan  the  third  side. 


PROPOSITION    VIII.     THEOREM. 

If  from  any  point  ivithin  a  triangle,  tico  straight  lines  he  drawn 
to  t/ie  ea:tremities  of  eitlier  side,  tlteir  sura  will  he  less  than 
that  of  the  two  'remaining  sides  of  the  triangle. 

Let  0  be  any  point  within  the  triangle  BAC,  and  let 
the  lines  OB,  DC,  be  di-aAvn  to  the  extremities  of  eitlier 
side,  as  BC]    then  will 

OB+OC<BA+Aa 

Let  BO  be  prolonged  till  it  meets  the 
side  AC  in  D     then 

OC<OI)+DC  (p.  7): 

add  BO  to  each,  and  we  have 

BO-i-OC<BO+OB+DC  {a.  4):  ^ 


BOOK    I.  27 

or,  BO-\-OC<BD^Da 

Bat,  BD<^BA+AD: 

add  DC  to  eacli,  and  we  have 

BD-^DC<:BA+AQ. 
But  it  has  been  shown  that 

BO+OG<:BD-\-DG: 
therefore,  still  more  is 

BO^OC<:BA-\-Aa 

PEOPOSITION  IX.     TIIEOEEM. 

If  two  triangles  have  two  sides  of  the  one  equal  to  two  sides  of 
the  other,  each  to  each,  and  the  included  anrjles  unequal,  the 
third  sides  ivill  he  unequal;  and  the  greater  side  will  belong 
to  the  triarigle  ivhich  has  the  greater  included  angle. 

Let  BAC  and  EDF  be  two  triangles,  having  the  side 
AB=DE,  AC^DF,  and  the  angle  A>D',  then  will  the 
side  BC  \)Q  greater  than  EF. 

Make  the  angle  CxlG=D',  take  AG=DE,  and  draw  GG. 

Then,  the  triangles  GAG  and  EDF  ^Y'A\  be  equal,  since 
the  J  have  two  sides  and  an  included  angle  in  each  equal, 
each  to  each  (p.  5) ;  consequently,  GG  is  equal  to  EF  (p.  5,  c). 

There  may  be  three  cases  in  this  proposition. 

1st.    When  the  point  G  falls  without  the  triangle  ^AC 

2d.    When  it  falls  on  the  side  BG\    and 

3d.    When  it  falls  within  the  triangle. 

Case  I.   In  the  triangles  AGG  and  ABG^  we  have, 

GI+IOGC-   and  A  ^ 

AI-{-IB>AB] 

therefore 

AG-hBG>GC+AB. 

Taking  away  AG 
from  the  one  side  and 
its   equal  AB  from   the   otl^er,  and   there   will   remain  BC 


28 


GEOMETRY. 


great(,'r  than  GC.  But  \ve  have  found  that  GC  \^  eijuiil  to 
EF  \    therefoi'e,  BG  will  be  greater  than  EF. 

Cdse  If.  If    the    point    G 

fall    on    the  side    BG^    it    is 

evident  that  GG^  or  its  equal 

EF,  will  be  shorter  than  BG 
(a.  8).  B  G 

Ga-^e  IT  J.  Lastly,    if  the    point    G 

fall  Avithin  the  triangle  BAG,  we 
shall  have 

AG-VGG<^AB^l]G, 

taking  AG  from  the  one,  and  its 
equal  AB  from  the  other,  there  will 
remain 

GG<BG  or  BG>EF. 

Cor.  Conversely:  if  two  sides  BA^ 
AG^  of  a  triangle  BAG,  are  equal  to 
two  sides  ED,  DF,  of  a  triangle  EDF, 
each  to  each,  while  the  third  side  BG 

of  the  first  is  greater  than  the  third  side  EF  of  the  second, 
then  the  ande  BAG  of  the  first  trianiile  will  be  oreater 
than  the  angle  EDF  of  the  second. 

For,  if  not  greater,  the  angle  BA  G  must  be  equal  to 
EDF  or  less  than  it.  In  the  tirst  case,  the  side  BG  would 
be  equal  to  EF  (p.  5,  c),  in  the  second,  BG  would  be  less 
than  EF ;  but  either  of  these  results  contrndicts  the 
hypothesis :   therefore,  BA  G  is  greater  than  EDF. 


TEOrOSITIOX    X.     THEOREM. 


If   two    triangles   have    the    iJtree    sides    of  the    ove  eqval  to  the. 
three  sides  of  the  other,  each  to  each,  the  triartgles  are  equal. 

Let  EDF  and  BAG  be  two  triangles,  having  the  sido 
ED=BA,  the  side  EF=BG,  and  the  "side  DF=AG\  then 
will  the  angle  D=A,  the  angle  E=B,  and  the  angle  F=C^ 
and  consequently  the  triangle  EDF  will  be  equal  to  the 
triangle  BAG. 


BOOK 


29 


For.  since  the  sid.^s 
ED^  J)F,  are  equal  to 
BA,  AC,  encl)  to  each,  if 
the  angle  I)  were  greater 
than  A,  it  would  follow, 
by  the  last  proposition, 
that  the  side  EF  would  be  greater  than  BC  \  and  if  the 
angle  D  were  less  than  A^  the  side  EF  would  be  less  than 
BC.  But  EF  is  equal  to  /iC,  b}^  hypothesis;  therefore, 
the  angle  D  can  neither  be  greater  nor  less  than  A  ;  there- 
fore it  must  be  equal  to  it.  In  the  same  manner  it  may 
be  shown  that  the  angle  E  is  equal  to  B^  and  the  angle 
F  io  C :    hence,  the  two  triangles  are  equal  (p.  6,  s). 

Scholium.  It  may  be  observed,  that  when  two  triangles 
are  equal  to  each  other,  the  equal  angles  lie  opposite  the 
equal  sides,  and  consequently,  the  equal  sides  opposite  the 
equal  angles :  thus,  the  equal  angles  D  and  A^  lie  opposite 
the  equal  sides  EF  and  BC. 


PROPOSITION  XI.     TIIEOEEM. 


In  an  isosceles  triangle,  the   angles   opposite   the   equal  sides  are 

equal. 

Let  BAC  be  an  isosceles  triangle,  having  the  side  BA 
equal  to  the  side  AG]  then  will  the  angle  0  be  equal  to 
the  angle  B. 

For,  join  the  vertex  J.,  and  the  mid- 
dle point  D,  of  the  base  BC.  Then,  the 
triangles  BAD^  JDAC,  will  have  all  the 
sides  of  the  one  equal  to  those  of  the 
other,  each  to  each.  For,  BA  is  equal  to  pj 
AC,  by  hypothesis,  AB  is  common,  and 
BB  is  equal  to  BC  by  construction :  therefore,  by  the  last 
proposition,  the  angle  B  is  equal  to  the  angle  C. 

Cbr.  1.  An  equilateral  triangle  is  likewise  equiangular, 
that  is  to  say,  has  all  its  angles  equal. 

Cor.  2  The  equality  of  the  triangles  BAB,  BAC,  proves 
iilso  that  the  angle  BAB,  is  equal  to  DAG^  and  BDA  to 


80  GEOMETEY. 

ABC\  hence,  the  latter  two  are  right  angles.  Therefore^ 
ike  line  draion  from  the  vertex  of  an  isosceles  triangle  to  the 
middle  point  of  the  lase,  divides  the  angle  at  the  vertex  into 
two  equal  parts,  and  is  peipendicular  to  tlie  base. 

Scholium.  In  a  triangle  which  is  not  isosceles,  any  side 
may  be  assumed  indifferently  as  the  base;  and  the  vertex 
is,  in  that  case,  the  vertex  of  the  opposite  angle.  In  an 
isosceles  triangle,  however,  that  side  is  generally  assumed 
as  the  base,  which  is  not  equal  to  either  of  the  other  two. 

PEOPOSITION  XII.     THEOKEM. 

Conversely:   If  two    angles   of  a    triangle   are   equal,  the   sides 
opposite  them  are  also  equal,  or,  the  triangle  is  isosceles. 

In  the  triangle  BAC^  let  the  angle  B  be  equal  to  the 
angle  A CB ;  then  will  the  side  AG  hQ  equal  to  the  side 
AB. 

For,  if  these  sides  are  not  equal,  sup- 
pose AB  to  be  the  greater.  Then,  take 
BD  equal  to  AC,  and  draw  CD.  N'ow, 
in  the  two  triangles  BDC^  BAC^  we  have 
BD  =A  CJ  by  construction ;  the  angle  B 
equal  to  the  angle  A  CB,  by  hypothesis ; 
and  the  side  BC  common:  therefore,  the 
two  triangles,  BDC,  BAC,  have  two  sides  and  the  included 
angle  of  the  one,  ^ual  to  two  sides  and  the  included 
angle  of  the  other,  t  ih.  to  each :  hence  they  are  equal 
(p.  5).  But  the  part  c^.  mot  be  equal  to  the  whole  (a.  8); 
hence,  there  is  no  ineqi  ility  between  the  sides  BA  and 
AC;   therefore,  the  triangle \^^'I(7  is  isosceles. 

PKOPOSITION   XIII.     THEOREM. 

Th^  greater  side  of  every  triangle  is  opposite  to  the  greater  angle} 
and  conversely,  the  greater  angle  is  opiposite  to  the  greater  side. 

First,  In  the  triangle  CAB,  let  the  angle  C  be  greater 
than  the  angle  B ;  then  will  the  side  AB,  opposite  (7,  be 
greater  than  AC,  opposite  B. 


BOOK    1 


31 


For,  make  the  angle  BCD=B. 
Then,  in  the  triangle  CDB^  we  shall 
have  CD=BD  (p.  12). 

Now,  the  side  AG<AD-\-DC] 

but  AD+DC=AD-\-DB=AB'. 

therefore    AG <^AB,  or,  AByAC. 

Secondly.  Suppose  the  side  ABy>AC \  then  will  the 
angle  (7,  opposite  to  AB^  be  greater  than  the  angle  B^ 
opposite  to  AC. 

For,  if  the  angle  C <^B^  it  follows,  from  what  has  just 
been  proved,  that  AB  <C^AG  \  which  is  contrary  to  the 
hypothesis.  If  the  angle  C—B^  then  the  side  AB—AG 
(p.  12) ;  which  is  also  contrary  to  the  supposition.  There- 
fore, when  AByAG^  the  angle  G  cannot  be  less  than  By 
nor  equal  to  it ;  therefore,  the  angle  G  must  be  greater 
than  B. 


PKOPOSITION    XIV.     THEOREM. 

Frora  a  given  point,  without  a  straight   line,  only  one  perpei^ 
dicular  can  he  drawn  to  that  line. 


Let  A  be  the  point,  and  DE  the  given  line. 

Let  us  suppose  that  we  can  draw 
two  perpendiculars,  AB^  AG.  Pro- 
long either  of  them,  as  AB^  till  BF 
is  equal  to  AB^  and  draw  FG.  Then 
the  two  triangles  GAB,  GBF,  will  be 
equal :  for,  the  angles  GBA  and  GBF 
are  right  angles,  the  side  GB  is  com- 
mon, and  the  side  AB  equal  to 
BFj  by  construction  ;  therefore,  the  two  triangles  are  equal, 
and  the  angle  AGB=BGF  (p.  5,  c).  But  the  angle  AGB 
is  a  right  angle,  by  hypothesis ;  therefore,  BGF  must  like 
wise  be  a  right  angle.  Now,  if  the  adjacent  angles  BGA^ 
BGF,  are  together  equal  to  two  right  angles,  AGF  must  be 
a  straight  line  (p.  3).  Whence,  it  follows,  that  between  the 
same  two  points,  A  and  F,  two  straight  lines  can  be 
drawn,    which   is   impossible   (a.  11) :    therefore,    only   one 


82 


GKOM  K'lliY 


perpendieiihir    can    be    drawn    IVoiii    llic    suine   })(>i!it    lo  the 
same  sti'aiglil  line. 

Cur.    At  a  ,uiven    point   C\    in   tlie  }^  jy 

line  A /^,  it  is  also  iinjx^ssible  to 
erect  nioi-e  than  one  perpendiealar  to 
that  line.     For,  if  6VJ,   CA]  were  both 

perjendieular     to     AB.     the     angles     a 

BCJJ,    BOA]     ^vould    both    be    right 

angles ;    henee,    they   would   be    ecpud  (a.  10),    and    a   part 

would,  be  equal  to  the  "whole,  whieh  is  impossible. 


PEOrOSITION  XV.     THEOREM. 

If  from  a  point  vMhout  a  straigld  line,  a  2ie7pendicidar  be  let 

fall  on   the   line,    and    ohlique   lines    he    drawn    to    different 

points: 
1st.    The  perpendicular  ivill  he  shorter  than  any  ohlique  line. 
*ld.    Any   two    ohlique    lines    ivhich   intersect    the  given    line   at 

points  equally  distant  from   the  foot  of  the   perptendicular^ 

will  he  equal. 
Hd.    Of  two  ohlique  lines  wJiich  intersect  the  given  line  at  poinl^s 

unequally  distant  from  the  pjerpendicular,  the  one  whdch  cuts 

off  the  greater  distance  will  he  the  longer. 

Let  A  be  the   given  point,  DE  the  given  line,  AB  the 
perpendicular,  and  AD,  AC,  AE,  the  oblique  lines. 

Prolong  the  perpendicular  AB  till 
BF  is  equal  to  AB,  and  draAV  FC, 
FD. 

First.  The  triangle  BCF,  is  equal 
to  the  triangle  CAB,  for  they  have 
the  right  angle  CBF=CBA,  the  side 
OB  common,  and  the  side  ^i'^^iM; 
hence,  the  third  sides,  OF  and  CA 
are  equal  (p.  5,  c).  But  ABF,  being  a  straight  line,  is 
shorter  than  ACF,  which  is  a  broken  line  (a.  12);  there- 
fore, "^5,  the  half  of  ABF,  is  shorter  than  AC,  the  half 
of  ACF;  hence,  the  perpendiculai'  is  shorter  than  any 
oblique  line. 


BOOK    L  33 

Secondly.  Let  us  suppose  BC=BE\  then  the  triangle 
CAB  will  be  equal  to  the  triangle  B2IE  ]  for  BC=BE,  the 
side  AB  is  common,  and  the  angle  CBA=ABE]]iQViCQ, 
the  sides  A  G  and  AE  ai^e  equal  (p.  5,  c) :  therefore,  two 
oblique  lines,  which  meet  the  given  line  at  equal  distances 
from  the  perpendicular,  are  equal. 

Thirdly.  Since  the  point  G  is  within  the  triangle  FDA^ 
the  sum  of  the  sides  FD,  I)A,  is  greater  than  the  sum  of 
the  lines  FG,  GA  (p.  8) :  therefore  AB,  the  half  of  the 
broken  line  FBA,  is  greater  than  AG,  the  half  of  EGA: 
consequently,  the  oblique  line  Avhicli  cuts  off  the  greater 
distance,  is  the  lon2:er. 

Gov.  1.  The  perpendicular  measures  the  shortest  distance 
of  a  point  from  a  line. 

Gor.  2.  From  the  same  point  to  the  same  straight  line, 
only  two  equal  straight  lines  can  be  drawn;  for,  if  there 
could  be  more,  we  should  have  at  least  two  equal  oblique 
lines  on  the  same  side  of  the  perpendicular,  which  is  im- 
possible. 


PROPOSITION  XVL     THEOREM. 

If  at  (he  middle  j^oi^it  of  a  given  straight  line,  a  j^erpendicular 
to  this  line  he  drawn: 

1st.   Any   2^oint    of  the.   jperjyendicular   vnll  he   equally    distant 

from  the  extremities  of  the  line: 
2d.   Any  point,  ivithout    the  perpendicular,   loill    he   unequally 

distant  from  the  extremities. 


Let  AB  be  the  given  straight  line,   G  its  middle 
and  EGF  the  perpendicular. 

First.  Let  D  be  any  point  of  the  per- 
pendicular; and  draw  BA  and  BE.  Then, 
since  A  (7=  (7-5,  the  two  oblique  lines  J.i), 
BB,  are  equal  (p.  15).  So,  likewise,  are 
the  two  oblique  lines,  AE,  EB,  the  two 
AF,  FB,  and  so  on.  Therefore,  any 
point  in  the  perpendicular  is  equally  dis- 
tant from  the  extremities  A  and  B. 

3 


pomt, 


/ 


34 


GEOMETRY, 


Secondly.  Let  /  be  any  point  out  of 
the  perpendicular.  If  lA  and  IB  be 
drawn,  one  of  these  lines  will  cut  the 
perpendicular  in  some  point  as  D ;  from 
this  point,  drawing  DB^  we  shall  have 
DB=DA.  But,  the  straight  hne  IB  is 
ess  than  IB+DB,  and 

ID+DB=ID-\-DA=IA ; 

therefore,  IB<.IA]  consequently,  any  point  out  of  the  per- 
pendicular, is  unequally  distant  from  the  extremities  A  and  B 

Cor.  Conversely:  if  a  straight  line  have  two  points  B 
and  F,  each  of  which  is  equally  distant  from  the  extremi- 
ties A  and  B,  it  will  be  perpendicular  to  AB  at  the  middle 
point  C. 


PEOPOSITION  XVII.     THEOEEM. 


//  tivo  riglit-angled  triangles  have  the  hypothenuse  and  a  side 
of  the  one  equal  to  the  hypothenuse  and  a  side  of  the  other^ 
each  to  each,  the  triangles  are  equal. 


Let  BAC  and  EDF  be  two  right-angled  triangles,  hav- 
mg  the  hypothenuse  AC=DF^  and  the  side  BA=ED:  then 
will  the  triangle  BAC  be  equal  to  the  triangle  EDF. 

If  the  sides  BO  and 
EF  are  equal,  the  tri-  ^ 
angles  are  equal  (p.  10). 
Now,  suppose  these  two 
sides  to  be  unequal,  and 
BC  to  be  the  greater. 

On  BC  take  BG=EF,  and  draw  AC.  Then,  in  the 
two  triangles  BAG^  EDF,  the  angles  B  and  E  are  equal, 
being  right  angles,  the  side  BA=ED  by  hjqDothesis,  and 
the  side  BG=EF  by  construction;  consequently,  AG=DF 
(p.  5,  c).  But  by  hypothesis  AG=DF\  and  therefore, 
AC=AG  (a.  1).  But  the  oblique  line  AC  cannot  be  equal 
\/o  AG,  since  BC  is  greater  than  BG  (p.  15)  ;  consequently, 
BO  and  EF  cannot  be  unequal,  and  hence,  the  triangles 
ar-c  equal  (p.  10). 


BOOK   I 


85 


PEOPOSITION   XVIII.     THEOKEM. 


If  two  straight  lines  are  perpendicular  to  a  third  line,  they  are 
parallel  to  each  other. 

Let   tlie   two  lines  AG^  BD^  be  perpendicular   io  AB^ 
then  will  they  be  parallel. 

For,  if  they  could  meet  in 
a  point  0,  on  either  side  of 
AB^  there  would  be  two  per- 
pendiculars OA^  OB^  let  fall 
from  the  same  point  on  the 
same  straight  line  ;  which  is 
impossible  (p.  14). 


B 


D 


:-::::^'0 


PROPOSITION  XIX.     THEOKEM. 


K- 


D 


If  two  straight  lines  meet  a  third  line,  maldng  the  sum  of  the 
interior  angles  on  the  sarae  side  equal  to  two  right  angles, 
the  two  lines  are  parallel. 

Let  the  two  lines  KC,  HD^  meet  the  line  BA^  making 
whe  angles  BAO^  ABD^  together  equal  to  two  right  angles: 
then  the  lines  KG^  HD^  Avill  be  parallel. 

From  G^  the  middle  point  of 
BA^  draw  the  straight  line  EGF^  H- 
perpendicular  to  KG  :  then,  it 
will  also  be  perpendicular  to  HD. 
For,  the  sum  BAG + ABB  is 
equal    to    two    right    angles,    by 

hypothesis ;  the  sum  ABD+ABE  is  likewise  equal  to  two 
right  angles  (p.  1) :  taking  away  ABD  from  both,  there 
will  remain  the  angle  BAG=ABE. 

Again,  the  angles  EGB^  AGE,  are  equal  (p.  4);  there- 
fore, the  triangles  EGB  and  AGE,  have  each  a  side  and 
two  adjacent  angles  equal  each  to  each ;  therefore  the  tri- 
angles are  equal,  and  the  angle  GEB  is  equal  to  GFA 
(p.  6,  c).  But  GEB  is  a  right  angle  by  construction; 
therefore,  GFA   is  a  right  angle  ;  hence,  the  two  lines  KC^ 


■C 


SQ  GEOMETRY. 

HD,  are  perpendicular  to  tlie  same  straight  line,  and  are 
tlierefore  parallel  (p.  18). 

Scholtum.  "Wlien  two  parallel  g 

straight  lines  AB^   CD^  are  met  / 

bj  a  third  line  FE^   the  angles         ^ G/ ^ 

which  are  formed  take  particu-  / 

lar  names.  / 

r^ J. J) 

Interior    angles    on    the    same  /H 

side,  are  those   which  lie  within  / 

the   parallels,  and  on   the   same 

side  of  the  secant  line ;'  thus,  HGB^  GHD^  are  interior 
angles  on  the  same  side;  and  so  also  are  the  angles  HGA, 
GHC. 

Alternate  angles  lie  within  the  parallels,  and  on  different 
sides  of  the  secant  line,  but  not  adjacent;  AGH^  GIID^  are 
alternate  angles ;    and  so  also  are  the  angles  GHC^  BGH. 

Alternate  exterior  angles  lie  without  the  parallels,  and  on 
different  sides  of  the  secant  hue,  but  not  adjacent :  EGB^ 
CHE,  are  alternate  exterior  angles;  so  also  are  the  angles 
AGE,  FED. 

Opposite  exterior  and  in'erior  angles  lie  on  the  same  side 
of  the  secant  line,  the  one  without  and  the  other  Avithin 
the  parallels,  but  not  adjacent:  thus,  EGB,  GED,  are 
opposite  exterior  and  interior  angles ;  and  so  also,  are  the 
angles  AGE,   GHC. 

Cor.  1.  If  two  straight  lines  meet  a  third  line,  making  the 
alternate  angles  equal,  the  straight  lines  are  xxirallel. 

Let  the  straight  hue  EF  meet  the  tvro  straight  lines  CD, 
AB,  making  the  alternate  angles  AGH,  GHD,  equal  to 
each  other:   then  will  AB   and  CD  be  parallel 

For,  to  each  of  the  equal 
angles,  add  the  angle  UGB;  we 
shall  then  have 

A  GH+HGB  =  GHD-^HGB. 

But  AGE+EGB  is  equal  to 
two  right  angles  (p.  1) :  hence, 
GHD+HGB  is  also  equal  to 
two  right  angles  (a.  1) :  then 
CD  and  AB  are  parallel  (p.  19.) 


BOOK    I.  37 

Cor.  2.  If  a  straight  line  EF^  meet  two  straight  lines 
CZ),  AB^  making  the  exterior  angle  EGB^  equal  to  the 
interior  and  opposite  angle  QHB^  the  two  lines  will  be 
parallel.  For,  to  each  add  the  angle  HGB:  we  shall  then 
have, 

EGB  -\-HGB  =  GHD+HGB : 

but  EGB  -\-EGB  is  equal  to  two  right  angles ;  hence, 
GHD+HGB  is  equal  to  two  right  angles ,  therefore,  CD, 
and  AB^  are  parallel  (p.  19). 


PKOrOSITION  XX.     TIIEOKEM. 

If  a  straight  line  meet  two  ^:)a?Y(?feZ  stravjht  lines,  the  sum  of 
the  interior  angles  on  the  same  side  ivill  he  equal  to  two 
right  angles. 

Let  the  parallels  AB^  CD^  be  met  bj  the  secant  line 
FE:  then  will  HGB  +  GHD,  or  IIGA-\-GHG,  be  equal  to 
two  right  angles. 

For,  if  HGB-VGHB  be 
not  equal  to  two  right  an- 
gles, let  IGB  be  drawn, 
making  the  sum  EGL  + 
GHB  equal  to  two  right  an- 
gles ;  then  IB  and  CD  will 
be  parallel  (p.  19);  and  hence, 
we  shall  have  two  lines  GB^ 
GB^  drawn  through  the  same  point  G  and  parallel  to  GB^ 
which  is  impossible  (a.  13) :  hence,  EGB  +  GHB  is  equal 
to  two  right  angles.  In  the  same  manner  it  may  be  proved 
that  HGA+GHC  is  equal  to  two  right  angles. 

Cor.  1.  If  HGB  is  a  right  angle,  GHB  will  be  a  right 
angle  also :  therefore,  every  straight  line  2^^y^^dicular  to  otuo 
of  two  parallels^  is  perpendicular  to  the  other. 

Cor.  2.  If  a  straight  line  meet  two  parallel  straight  lines,  the 
aliefrnate  angles  tuill  he  equal. 

Let  ABj  CD,  be  two  parallels,  and   FE  the  secant  line. 


38  GEOMETKY 

The    sum    HGB  +  GHD    is 
al  to  two  right  angles.      But 
the   sum  HGB+HGA    is    also 


equal  to  two  right  angles.      But  / 


equal  to  two   right  angles  (p.  1).  Y 

Taking  from  each  the  angle  HGB^  / 

and  there  remains  AGH=GHD.    C  -g  D 

In    the    same    manner    we    may  / 

prove  that  GHC=HGB.  F 

Cor.  3.  If  a  straight  line  meet  two  parallel  lines,  the  opjDO- 
site  exterior  and  interior  angles  will  be  equal.  For,  the  sum 
HGB  -\-  GHD  is  equal  to  two  right  angles.  But  the  sum 
HGB-\-EGB  is  also  equal  to  two  right  angles.  Taking  from 
each  the  angle  HGB,  and  there  remains  GHD=EGB.  In  th^ 
same  manner  we  may  prove  that  GHG=AGE. 

Scholium.  TVe  see  that  of  the  eight  angles  formed  by  a 
line  cutting  two  parallel  lines  obliquely,  the  four  acute 
angles  are  equal  to  each  other,  and  so  also  are  the  four 
obtuse  angles. 


FKOPOSITIOX  XXI.     THEOREM. 

If  two  straight  lines  meet  a  third  line,  maldng  the  sum  of  the 
interior  angles  on  the  saine  side  less  than  two  right  angles^ 
the  two  lines  icill  meet  if  sufficientlg  produced. 

Let  the  two  lines  CD,  IL,  meet  the  line  EF,  making  the 
sum  of  the  interior  angles  HGL,  GHD,  less  than  two  right 
angles  :   then  will  IL  and  CD  meet  if  sufficiently  produced. 

For,  if  they  do  not  meet 
they  are  parallel  (d.  16).  But 
they  are  not  parallel,  for  if 
they  were,  the  sum  of  the 
interior  angles  LGH,  GHD, 
would  be  equal  to  two  right 
angles  (p.  20),  whereas  it  is 
less  by  hypothesis :  hence,  the 
lines  IL,   CD,  will  meet  if  sufficiently   produced. 

Cor.  It  is  evident  that  the  two  lines  IL,  CD,  will  meet 
on  that  side  of  EF  on  which  the  sum  of  the  two  .angles 
HGL,  GHDj  is  less  than  two  right  angles. 


BOOK    I.  39 


PEOPOSITION  XXII.     THEOREM. 

Two  straight   lines   which   are  parallel  to  a  third  line,  are 
parallel  to  each  other. 

Let  CD  and  AB  be  parallel  to  the  third  line  EF ;  theu 
are  they  parallel  to  each  other. 

Draw  PQR  perpendicular  to  EF^      ^ 

and   cutting  AB^   CD^    in  the  points 

P  and  Q.     Since  AB  is   parallel  to      C — 

EF^  PR  will  be  perpendicular  to  AB        

(p.  20,  c.  1) ;   and  since  CD  is  parallel 

to  EF^  PR  will  for  a  like  reason  be 

perpendicular  to  CD.     Hence,  AB  and  CD  are  perpendicular 

to  the  same  straight   line  ;    hence,  they  are  parallel  (p.  18). 


D 


Q 
B 


PROPOSITION  XXIII.  THEOREM. 
Two  parallels  are  everywhere  equally  distant. 

Let  CD  and  AB  be  two  parallel  straight  lines.  Through 
any  two  points  of  AB,  as  F  and  E,  suppose  FH  and  EG 
to  be  drawn  perpendicular  to  AB.  These  lines  will  also 
be  perpendicular  to  CD  (p.  20,  C  1)  ;  and  we  are  now  to 
show  that  they  will  be  equal  to  each  other. 

If  GF  be  drawn,  the 
angles  GFE,  FGH,  consid- 
ered in  reference  to  the  par- 
allels AB,  CD,  will  be  alter- 
nate angles,  and  therefore, 
equal  to  each  other  (p.  20,  c.  2).  Also,  the  straight  lines 
FH,  EG,  being  perpendicular  to  the  same  straight  line  AB, 
are  parallel  (p.  18) ;  and  the  angles  EGF,  GFH,  considered 
in  reference  to  the  parallels  FH,  EG,  will  be  alternate 
angles,  and  therefore  equal.  Hence,  the  two  triangles  EFG^ 
FGH,  have  a  common  side,  and  two  adjacent  angles  in 
each  equal ;  therefore,  the  triangles  are  equal  (p.  6) ;  conse- 
quently, FH,  which  measures  the  distance  of  the  parallels 
AB  and  CD  at  the  point  F,  is  equal  to  EG,  which  mea- 
sures the  distance  of  the  same  parallels  at  the  point  E. 


40  GEOMETRY 


PKOPOSITION    XXIV.     THEOREM. 

Tf  tv:o  angles   have  their  sides  parallel  and   lying  in  the  same 
direction,  they  ivill  he  equal. 

Let  BAC  and  DEF  be  the  two  angles,  having  AB 
parallel    to    ED^  and  AG  io  EF ]   then  will   they  be  equal. 

For,  produce  BE^  if  necessary,  till  p        D 

It   meets   AG  in  G.      Then,   since  EF  /   v/ 

is   parallel   to   GG,  the   angle  BEE  is  /    ^[         ^ 

equal  io  BGG  (p.  20,  c.  3);    and  since  -^      r C 

BG  i^  parallel  to  AB,  the  angle  BGG  H ^ F 

is   equal   to   BAG \    hence,    the   angle 
VEF  is  equal  to  BAG  (a.  1). 

Scholium.  The  restriction  of  this  proposition  to  the  case 
where  the  side  EF  lies  in  the  same  direction  with  AG, 
and  EB  in  the  same  direction  with  AB,  is  necessary, 
because  if  FE  were  prolonged  towards  H,  the  angle  BEH 
would  have  its  sides  parallel  to  those  of  the  angle  BAG, 
but  would  not  be  equal  to  it.  In  that  case,  BEH  and 
BAG  would  be  together  equal  to  two  right  angles.  For, 
BEH -{-BEE  is  equal  to  two  right  angles  (p.  1);  but  BEF 
is  equal  to  BAG:  hence,  BEH+BAG  is  equal  to  two 
right  angles. 


PEOPOSITION  XXV.     TIIEOEEM. 

f)i    every    triangle  the  sura   of  the  three  angles  is  equal  to  two 
■right  angles. 

Let   ABG  be  any  triangle :   then    will    the    sum  of  the 
angles  G+A+B  be  equal  to  two  right  angles. 

For,  prolong  the  side  GA  towards 
D,  and  at  the  point  A,  suppose  AE 
to  be  drawn,  parallel  to  BG.  Then, 
since  AE,  GB,  are  parallel,  and  GAB 
cuts  them,  the  exterior  ans^le  BAE 
is  equal  to  its  interior  opposite  angle  G  (p.  20,  c.  3).  In 
like  manner,  since  AE,  GB,  are  parallel,  and  AB  cuts  them. 


BOOK    I.  41 

tlie  alternate  angles  B  and  BAE^  arc  equal;  hence,  the 
three  angles  of  the  triangle  BAG  are  equal  to  the  three 
angles  CAB^  BAF,  BAD,  each  to  each;  but  the  sum  of 
these  three  angles  is  equal  to  t^YO  right  angles  (p.  1) ;  con- 
sequently, the  sum  of  the  three  angles  of  the  triangle,  is 
equal  to  two  right  angles  (A.  1). 

Cor.  1.  Two  angles  of  a  triangle  being  given,  or  mere- 
ly their  sum,  the  third  will  be  found  by  subtracting  that 
sum  from  two  right  angles. 

Cor.  2.  K  two  angles  of  one  triangle  are  respectively 
equal  to  two  angles  of  another,  the  third  angles  will  also 
be  equal,  and  the  two  triangles  will  be  mutually  equian- 
gular. 

Cor.  3.  In  any  triangle  there  can  be  but  one  right 
angle :  for  if  there  were  two,  the  third  angle  must  be 
nothing.      Still   less,  can    a  triangle   have    more   than    one 

obtuse  angle. 

Cor.  4.  In  every  right-angled  triangle,  the  sum  of  the 
two  acute  angles  is  equal  to  one  right  angle. 

Cor.  6.    Since   every  equilateral  triangle   is  also   equian 
gular  (p.  11,  c.  1),  each  of  its  angles  will   be   equal  to   the 
third  part  of  two  right  angles ;    so,  that,  if  the  right  angle 
is  expressed  by  unity,  each  angle  of  an  equilateral  triangle 
will  be  expressed  by  f . 

Cor.  6.  In  every  triangle  ABC,  the  exterior  angle  BAB 
is  equal  to  the  sum  of  the  two  interior  opposite  angles  B 
and  C  For,  AB  being  parallel  to  BC,  the  part  BAE  is 
equal  to  the  angle  B,  and  the  other  part  BAE  is  equal  to 
the  angle  C. 


PEOPOSTTION    XXVI.     TIIEORE^I. 

The  sum  of  all   the   interior   angles   of  a  polygon,  is  equal  to 
tivice  as  many  right  angles,  less  four,  as  the  figure  has  sides. 

Let  ABCDE  be  any  polygon :   then  will  the  sum  of  its 
interior  angles 


42  GEOMETEY. 

A-\-B-\-C-VD+E 
be   equal  to   twice   as  many  riglit   angles,  less  four,  as  the 
figure  lias  sides. 

From  the  vertex  of  any  angle  J., 
draw  diagonals  AC^  AD,  to  the  ver- 
tices  of  the  other  angles.  It  is  plain 
that  the  polygon  will  be  divided 
into   as   many   triangles,    less   two,   as 

it  has  sides;   for,  these   triangles  may  

be   considered   as  ha^T.ng  the  point  ^  A  B 

for  a  common  vertex,  and  for  bases,  the  several  sides  of 
the  polygon,  excepting  the  two  sides  which  form  the  angle 
A.  It  is  evident,  also,  that  the  simi  of  all  the  angles  in 
these  triangles  does  not  differ  from  the  sum  of  all  the 
angles  in  the  polygon:  hence,  the  sum  of  all  the  angles 
of  the  polygon  is  equal  to  two  right  angles,  taken  as 
many  times  as  there  are  triangles  in  the  figure ;  that  is,  as 
many  times  as  there  are  sides,  less  two.  But  this  pro- 
duct is  eqnal  to  t^T.ce  as  many  right  angles  as  the  figure 
has  sides,  less  four  right  angles. 

Cor.  1.  The  sum  of  the  interior  angles  in  a  quad- 
rilateral is  equal  to  two  right  angles  multiplied  by  4—2, 
which  amounts  to  four  right  angles :  hence,  if  all  the 
angles  of  a  quadrilateral  are  equal,  each  of  them  will  be  a 
right  angle.  Hence,  each  of  the  angles  of  a  rectangle,  and 
of  a  square,  is  a  right  angle  (d.  25). 

Cor.  2.  The  sum  of  the  interior  angles  of  a  pentagon 
is  equal  to  two  right  angles  multiphed  by  5—2,  which 
amounts  to  six  riglit  angles  :  hence,  when  a  pentagon  is 
equiangular,  each  angle  is  equal  to  the  fifth  part  of  six 
right  angles,  or  to  f  of  one  right  angle. 

Cor.  8.  The  sum  of  the  interior  angles  of  a  hexagon  is 
equal  to  2X  (6— 2,)  or  eight  right  angles;  hence,  in  the 
equiangular  hexagon,  each  angle  is  the  sixth  part  of  eight 
right  angles,  or  ^  of  one. 

Cor.  4.  In  any  equiangular  polygon,  any  interior  angle 
is  equal  to  t^dce  as  many  right  angles,  less  four, -as  the 
figure  has  sides,  divided  by  the  number  of  angles. 


BOOK   I. 


43 


Scholium.   When  this  proposition  is  applied 
to  polygons  which  have  re-entrant  angles,  each 
re-entrant  angle   must   be   regarded  as  greater 
than  two  right  angles.     But  to  avoid  all  ambi- 
guity, we  shall  henceforth  limit  our  reasoning 
to   polygons   with   salient  angles,    which    are   named  convex 
polygons.     Every  convex   polygon    is    such,   that    a    straight 
line,  drawn  at  pleasure,  cannot  meet  the  sides  of  the  poly- 
gon in  more  than  two  points. 


PKOPOSITION  XXVII.     THEOKEM. 

If  the  sides  of  any  polygon  he  prolonged,  in  the  same  direction, 
the  sum  of  the  exterior  angles  will  he  equal  to  four  rigid 
angles. 

Let  the  sides  of  the  polygon  ABCDFG^  be  prolonged,  in 
the  same  direction ;  then  will  the  sum  of  the  exterior  angles 

a-\-h-\-c-\-d  +  f-\-g, 

be  equal  to  four  right  angles. 

For,  each  interior  angle,  plus  its 
exterior  angle,  as  A+a^  is  equal  to 
two  right  angles  (p.  1).  But  there 
are  as  many  exterior  as  interior 
angles,  and  as  many  of  each  as 
there  are  sides  of  the  polygon : 
hence  the  sum  of  all  the  interior 
and  exterior  angles,  is  equal  to  twice  as  many  right  angles 
as  the  polygon  has  sides.  Again,  the  sum  of  all  the  inte- 
rior angles  is  equal  to  twice  as  many  right  angles  as  the 
figure  has  sides,  less  four  right  angles  (p.  26).  Hence,  the 
interior  angles  plus  four  right  angles,  is  equal  to  tAvice  as 
many  right  angles  as  the  polygon  has  sides,  and  conse- 
quently, equal  to  the  sum  of  the  interior  angles  plus  the 
sum  of  the  exterior  angles.  Taking  from  each  the  sum  of 
the  interior  angles,  and  there  remains  the  sum  of  the  exte- 
rior angles,  equal  to  four  right  angles. 


44  GEOMETEY. 

PROPOSITION^'  XXYIII.    THEOREM. 

In  every  2^ciJ'ci^^eIogra7n,  the  oirposite  sides  and  angles  are  equal 
each  to  each. 

Let  ABCD  be  a  parallelogram:  tlien  will  AB=DCj 
AJD=BG,  the  angle  A=C,  and  tlie  angle  ADC=ABO. 

For,  draw  tlie  diagonal  BB^  dividing 
tlie  parallelogTam  into  the  tv\'o  trian- 
gles, ABB,  BBC.  Kow,  since  AB,  BC, 
are  parallel,  the  angle  ABB = BBC  (p. 
20,  c.  2) ;  and  since  AB,  CB,  are  parallel, 
the  angle  ABB= BBC :  and  since  the 

side  BB  is  common,  the  two  triangles  are  equal  (p.  6); 
therefore,  the  side  AB^  opposite  the  angle  ABB,  is  equal 
to  the  side  BC^  opposite  the  equal  angle  BBC  (p.  10,  s.), 
and  the  third  sides  AB^  BC,  are  equal :  hence,  the  oppo- 
site sides  of  a  parallelogram  are  equal. 

Again,  since  the  triangles  are  equal,  the  angle  A  is 
equal  to  the  angle  C  (p.  10,  s.)  Also,  the  angle  ABC  com- 
posed of  the  two  angles,  ABB,  BBC,  is  equal  to  ABC, 
composed  of  the  correspondiDg  equal  angles  BBC,  ABB 
(a.  2) :  hence,  the  opposite  angles  of  a  parallelogram  are  equal. 

Cor.  1.  Two  parallels  AB,  CB,  included  between  two 
other  parallels  AB,  BC,  are  equal :  and  the .  diagonal  BB 
divides  the  parallelogram  into  two  equal  triangles. 

Cor.  2.  Two  parallelogTams  which  have  two  sides  and 
the  included  angle  in  the  one  equal  to  two.  sides  and  the 
included  angle  in  the  other,  each  to  each,  are  equal. 

Let  the  parallelogram  ABCB,  have 
the  sides  AB,  AB,  and  the  included 
angle  BAB  equal  to  the  sides  AB,  AB, 
and  the  included  angle  BAB,  in  the 
next  figure;  then  will  they  be  equal. 

For,  in  each  figure,  draw  the  diagonal  BB.  By  the  last 
corollary,  the  diagonal  divides  each  parallelogram  into  two 
equal  triangles :  but  the  triangle  BAB  in  one  parallelo- 
gram, is  equal  to  the  triangle  BAB  in  the  other  (p.  6) : 
hence,  the  parallelograms  are  equal  (a.  6). 


BOOK    I.  45 


PKOPOSITION  XXIX.     TIIEOEEM. 

If  the  opj)osite  sides  of  a  quadrilateral  are  equals  each  to  each^ 
the  equal  sides  are  ]oarallel^  and  the  figure  is  a  parallelogram. 

Let  ABCD  be  a  quadrilateral,  having  its  opposite  sides 
respectively  equal,  viz.:  AB=DO,  and  AjD=BO;  then  will 
these  sides  be  parallel,  and  the  figure  a  parallelogTam. 

For,  ha  vino;  drawn  the  diao'onal  BB 
the  two  triangles  ABB,  BBC,  have  all 
the  sides  of  the  one  equal  to  the  cor- 
responding sides  of  the  other ;  there- 
fore thej  are  equal,  and  the  angle  ABB, 
opposite  the  side  AB,  is  equal  to  BBC,  opposite  CB  (p.  10, 
s.) ;  therefore  the  side  AB  is  parallel  to  BG  (p.  19,  c.  1) 
For  a  hke  reason  AB  is  parallel  to  CB :  therefore,  the 
quadrilateral  ABCB  is  a  parallelogram. 


PEOPOSITION  XXX.     THEOKEM. 

If  two  op2)osite  sides  of  a  quadrilateral  are  equal  and  2iarallel, 
the  other  sides  are  equal  and  parallel,  and  the  figure  is  a 
parallelogram. 

Let  ABCB  be  a  quadrilateral,  having  the  sides  AB, 
CB,  equal  and  parallel ;  then  will  the  figure  be  a  parallel- 
ogram. 

For,  draw  the  diagonal  BB,  divid- 
ing the  quadrilateral  into  two  trian- 
gles. Then,  since  AB  is  parallel  to 
BC,  the  alternate  angles  ABB,  BBC 
are  equal  (p.  20,  c.  2) ;  moreover,  the 
side  BB  is  common,  and  the  side  AB=BC;  hence,  the 
triangle  ABB  is  equal  to  the  triangle  BBC  (p.  5) ;  there- 
fore, the  side  AB  is  equal  to  BC,  the  angle  ABB=BBC, 
and  consequently  AB  is  parallel  to  BC  (p.  19,  c.  1) ;  hence, 
the  figure  ABCB  is  a  parallelogram. 


46 


GEOMETRY. 


PKOPOSITION  XXXI.     THEOEEM. 


TJte    two    diagonals  of  a  parallelogram   divide  each   otJier  into 
equal  parts,  or  mutually  bisect  each  other. 

Let  ADCB  be  a  parallelogram,  AC  and  DB  its  diago 
nals,   intersecting    at    E\  then   ^ill    AE=ECj    and    DE= 

EB. 

Comparing  the  triangles  AED^  BEC^ 
we  find  the'^side  AD=CB  (p.  28),  the 
angle  ADB  =  CBE,  and  the  angle 
DAE=ECB  (p.  20,  c.  2) ;  hence,  these 
triangles  are  equal  (p.  6);  consequently, 
AE^  the  side  opposite  the  angle  ADE^  is  equal  to  EC, 
opposite  CBE,  and  DE  opposite  DAE  is  equal  to  EB 
opposite  ECB. 

Scholium.  In  the  case  of  the  rhombus,  the  sides  AB^ 
BCj  being  equal,  the  triangles  AEB,  EBC,  have  all  the 
sides  of  the  one  equal  to  the  corresponding  sides  of  the 
other,  and  are  therefore  equal :  whence,  it  follows,  that  the 
angles  AEB,  BEC,  are  equal,  and  therefore,  the  two  diago- 
nals of  a  rhombus  bisect  each  other  at  right  angles. 


BOOK    II. 

OF    RATIOS    AND    PROPORTIONS, 


DEFINITIONS. 


1.  Proportion  is  the  relation  whicli  one  magnitude 
bears  to  another  magnitude  of  the  same  kind,  with  respect 
to  its  being  greater  or  less.* 

2.  Ratio  is  the  measure  of  the  proportion  which  one 
magnitude  bears  to  another;  and  is  the  quotient  which 
arises  from  dividing  the  second  by  the  first.  Thus,  if  A 
and  B  represent  magnitudes  of  the  same  kind,  the  ratio 
of  A  to  B  is  expressed  by 

B 

A  and  B  are    called    the   terms    of   the    ratio ;   the  first  is 
called   the   antecedent^  and  the  second,  the  consequent. 


3.  The  ratio  of  magnitudes  may  be  expressed  by  num- 
bers, either  exactly  or  approximatively ;  and  in  the  latter 
case,  the  approximation  may  be  brought  nearer  to  the  true 
ratio  than  any  assignable  difference. 

Thus,  of  two  magnitudes,  one  may  be  considered  to 
be  divided  into  some  number  of  equal  parts,  each  of  the 
same  kind  as  the  whole,  and  regarding  one  of  these 
parts  as  a  unit  of  measure,  the  magnitude  may  be  expressed 
by  the  number  of  units  it  contains.  If  the  other  magni- 
tude contain  an  exact  number  of  these  units,  it  also  may 

*  See  Duvies'  Logic  of  Mathematics :   Proportion,  §  267. 


48  GEOMETKY. 

be    expressed   bj   tlie   number  of  its    units,    and   the    two 
magnitudes  are  then  said  to  be  commensurable. 

If  the  second  magnitude  do  'not  contain  the  measuring 
unit  an  exact  number  of  times,  there  maj  perhaps  be  a 
smaller  unit  ^vhich  will  be  contained  an  exact  number  of 
times  in  each  of  the  magnitudes.  But  if  there  is  no  unit 
of  an  assignable  value,  which  is  contained  an  exact  number 
of  times  in  each  of  the  magnitudes,  the  magnitudes  arc 
said  to  be  incommensurable. 

It  is  plam,  however,  that  if  the  unit  of  measure  bt 
repeated  as  many  times  as  it  is  contained  in  the  second 
magnitude,  the  result  ^x^\  differ  from  the  second  magni- 
tude b}^  a  quantity  less  than  the  unit  of  measure,  since 
the  remainder  is  always  less  than  the  divisor.  jSTow,  since 
the  unit  of  measure  may  be  made  as  small  as  we  please, 
it  follows,  that  magnitudes  may  be  represented  by  num- 
bers to  any  degree  of  exactness,  or  they  will  differ  from 
their  numerical  representatives  by  less  than  any  assign- 
able magnitude. 

4.  TTe  will  illustrate  these  principles  by  finding  the 
ratio  between  the  straight  lines  CD  and  AB^  which  we  will 
suppose  commensurable. 

From  the  gi'eater  Hne  AB^  cut  off  a  part  equal    A    C 
to  the  less  CD^  as  many  ti];nes  as  possible ;  for  ex- 
ample, twice,  with  the  remainder  BE. 

From  the  line  CD^  cut  off  a  part,  CF^  equal  to 
the  remainder  BE^  as  many  times  as  possible ; 
once,  for  example,  with  the  remainder  DF. 

From   the   first    remainder  BE^  cut    off  a  part 
equal   to    the   second,  DF^  as  many  times  as  possi-    Ig 
ble ;   once,  for  example,  wiih.  the  remainder  BG. 

From  the  second  remainder  DF^  cut  off  a  part 
equal  to  BG^  the   third  remainder,  as   many  times    ^ 
as  possible. 

Continue  this  process,  till  a  remainder  occurs,  which  is 
contained  exactly,  a  certain  number  of  times,  in  the  pre- 
ceding one. 

Then,  this  last  lemainder  will  be  the  common  measure 
of  the  proposed  lines.      Eegarding   this   as   unity,  we  shall 


"G 


BOOK    II.  49 

easily  find  the  values  of  the  preceding  remainders;  and  at 
last,  those  of  the  two  proposed  hues,  and  hence,  their  ratio 
in  nnnibers. 

Sup|)ose,  for  instance,  we  find  GB  to  be  contained 
exactly  twice  in  FD]  BG  will  be  the  common  measure  of 
the  two  proposed  lines.  Put  I>G=1]  we  shall  then  have, 
FI)=^2 ;  but  BB  contains  FB  once,  j^^^^  ^^>  tlierelbre, 
we  have  EIJ  =  S:  CB  contains  E8  once,  2)lus  FB  ;  there- 
fore, we  have  CB=d  :  and  lastly,  AB  contains  CB  twice, 
plus  FB  ]  therefore,  we  have  AB=VS;  hence,  the  ratio  of 
the  lines  is  that  of  5  to  13.  If  the  line  CD  were  taken 
for  unity,  the  line  AB  would  be  ^/  ;  if  AB  were  taken 
for  unity,   CB  would  be  j^. 

5.  "What  has  been  shown,  in  respect  to  the  straight 
lines,  CB  and  ABj  is  equally  true  of  any  two  magnitudes, 
A  and  B. 

For,  we  may  conceive  A  to  be  divided  into  a  number  J/ 
of  units,  each  equal  to  A':  then  A=MxA':  let  B  be 
divided  into  a  number  H  of  equal  units,  each  equal  to  J.' ; 
then  B=NxA' ;  M  and  N  being  integer  numbers.  Now 
the  ratio  of  A  to  i?,  will  be  the  same  as  the  ratio  of 
MxA'  to  NxA' \  that  is,  the  same  as  the  ratio  of  the 
numerical  quantities  M  and  iVJ  since  A'  is  a  common  unit.  ' 

6.  If  there  be  four  magnitudes,  A^  B,  Cj  and  B,  having 
such  values  that 

B_B 
A~C' 

then  A  is  said  to  have  the  same  ratio  to  B,  that  C  has  to 
B;  or,  the  ratio  of  J.  to  ^  is  said  to  be  equal  to  the 
ratio  of  C  to  B.  When  four  quantities  have  this  relation 
to  each  other,  they  are  said  to  be  in  proportion. 

To  indicate  that  the  ratio  of  J.  to  jS  is  equal  to  the 
ratio  of  C  to  B,  the  quantities  are  usually  written  thus, 

A  :  B:  :  C  :  B, 

and  read,  J.  is  to  ^  as  (7  is  to  B.  The  quantities  which 
are  compared  together  are  called  the  terms  of  the  propor- 
tion. The  first  and  last  terms  are  called  the  two  extremes^ 
and  the  second  and  third  terms,  the  two  means, 

4 


50  GEOMETEY. 

7.  Of  four  proportional  quantities,  the  last  is  said  to  he 
a  fourth  proportional  to  the  other  three,  taken  in  order. 
The  first  and  second  terms,  are  called  the  first  couplet  of 
the  proportion ;  and  the  third  and  fourth  terms,  the  second 
amplet:  the  first  and  third  terms  are  called  the  antecedents^ 
nnd  tlie  second  and  fourth  terms,  the  consequents. 

8.  Three  quantities  are  in  proportion,  Trhen  the  first 
has  the  same  ratio  to  the  second,  that  the  second  has  to 
the  third ;  and  then  the  middle  term  is  said  to  be  a  mean 
propjortional  between  the  other  two. 

9.  Magnitudes  are  in  proportion  by  alternation^  or  alter- 
nately, when  antecedent  is  compared  with  antecedent,  and 
consequent  with  consequent. 

10.  Magnitudes  are  in  proportion  by  inversion^  or  2*n- 
^'ersel^J^  when  the  consequents  are  taken  as  antecedents,  and 
the  antecedents  as  consequents. 

11.  Mag-nitudes  are  in  proportion  by  composition^  when 
the  sum  of  the  antecedent  and  consequent  is  com.pared 
either  with  antecedent  or  consequent. 

12.  Magnitudes  are  in  proportion  by  division^  when  the 
difference  of  the  antecedent  and  consequent  is  compared 
either  with  antecedent  or  consequent. 

13.  Equimultiples  of  two  quantities  are  the  products 
which  arise  from  multiplying  the  quantities  by  the  same 
numbei  :  thus,  ?72X-1,  mXB^  are  equimultiples  of  A  and  B^ 
the  common  multiplier  being  vu 

14.  Two  varying  quantities,  A  and  B^  are  said  to  be 
reciprocally  propjortional^  or  inversely  j^'^'oportional^  when  their 
values  are  so  changed  that  one  is  increased  as  many  times 
as  the  other  is  diminished.  In  such  case,  either  of  them 
is  alTrays  equal  to  a  constant  quantity  divided  by  the 
other,  and  their  product  is  constant. 


BOOK    II.  51 


PKOPOSITION  L     THEOREM. 

When  four   magnitudes   are   in  proportion,  the  product  of  iht 
two  extremes  is  equal  to  the  product  of  the  two  means. 

Let  J.,  B,  C,  D,  be  any  four  magnitudes,  and  i/j  iV,  P, 
ft  their  numerical  representatives ; 

then,  if  21    :     N    :  :     F    :     Q, 

we  shall  have  MxQ=NxR 

For,  since    the   magnitudes   are  in    proportion,   we  have 
(D.  6), 

_  =  ^;  therefore, 

Q 
N=MXp',    whence,    NxP=MxQ. 

Cor.  If  there  are  three  proportional  quantities,  the 
product  of  the  extremes  will  be  equal  to  the  square  of 
the  mean  (d.  8).     For,  if  N=P,  we  have 

i/X  5=A^"  or  P^ 


PEOrOSITION  II.     THEOEEM. 

If  the  product  of  two  magnitudes  he  equal  to  the  product  of 
two  other  magnitudes^  two  of  them  may  he  made  ilie  ex- 
tremes a7id  the  other  two  the  means  of  a  proportion. 

If  we  have  Mx  Q=NxP\  then  will  M :  N  .  :  P  :   Q. 

For,  if  P  have  not  to  §,  the  ratio  which  M  has  to  iVJ 
let  P  have  to  Q\  (a  number  greater  or  less  than  ft)  the 
same  ratio  which  M  has  to  N:   that  is,  let 

M    :     N    :  :     P    :     ft'  ; 
then  (p.  1),  MxQ'=NxP\ 

,  ^,      NXP  .    ^        ^     NxP 

hence,  ft  =—jf-  »   but,   ft=  —^ — : 


Consequentl}^,  ()'=  ft  and  the  supposition  that  it  is  either 
greater  or  less,  is  absurd;  hence,  the  four  magnitudes  J/, 
iV;  P,   ft,  are  proportional. 


62  GEOMETRY. 


rKOPOSITION    III.     THEOREM, 

Jf  four  magnitudes    are   in   proportion^    tliey  will  he   in   pro- 
portion wlien   taken   alternately. 

Let  J/J  X^  P,  (),  be  four  quantities  in  proportion ;  so  that 
M  '.   X  ::   P  :    Q]   then   will  M  :   P  :  :   X  ;    Q. 
For,  since   21   :    X    :  :    P  :    Q:    we   have  MxQ=XxP] 
therefore  J/  and   Q  may  be  made  the  extremes,  and  X  and 
P  the  means  of  a  proportion  (p.  2) ; 

hence,  II    :    P    :  :     X    :     Q. 

PEOPOSITION   IV.     THEOREM. 

If  there  he  four  proportional  magnitudes,  and  four  other  pro- 
portional magnitudes,  having  the  antecedents  tJie  same  in 
hothj  Uie  consequents  will  he  proportional. 


Let     M  '.    X 
and  M  :    R 

thenwill.Y   :     Q 


P  :  Q,  giving  J/x  Q  =  XxP, 
P  :  S,  giving  RxP=MxS, 
R    :    S. 

For,  multiplying  the  equations  member  by  member, 
MxQxRxP=MxSxXxP\ 
canceUing  MxP  in  both  members,  we  have, 
QxR=SxX:   hence  (p.  2), 
X    '.     Q    :  :     R    :     S. 

Cor.  K  there  be  two  sets  of  proportionals,  in  which 
the  ratio  of  an  antecedent  and  consequent  of  the  one  is 
equal  to  the  ratio  of  an  antecedent  and  consequent  of  the 
other,  the  remaining  terms  will  be  proportional. 

For,  if  Ave  had  the  two  proportions, 

M    :     P    :  :     X    :     Q   and   R    :     S    :  :     T    :      V, 
we  shall  also  have 

M     X  R     T 

XT        -o    P       S    ^.        Q       V 
^""'  '^    M^-R^    ^^'"   .Y=T' 
aiid  we  shall  have     X    :     Q    :  :     T    \     V, 


BOOK    II.  53 

PROPOSITION    V.     THEOREM. 

[f  four  magnitiales  are  in  proportions  they  will  ht  in  proportion 
wlitn  talctn  inversdij. 

If  M  :  N  :  :  P  :  Q,  then  will  K  :  M  :  :  Q  :  P. 
For,  from  tlie  given  })r(jp()iti()U,   we  have 

MxQ  =  Nxl\    or,  NxP  =  JTxQ. 

Now,  JS^  and  P  may  be  made  ihe  extremes,  and  If  and   Q 
the  TTieans  of  a  ])ru[H)rti()n  (J*.  2j :    hence 

N    :     M    :  :     Q     :     P. 

PROPOSITION    VI.     THKOKEM. 

If  four  magnitudes   are    in    projtortiini,  tJiey  will  he  in  propor- 
tion   by  coiiijiosUion  or  divi^^ion. 

If  we  have        J/    :     N  -.  -.   P    '.     ft 

we  shall  also  have  M^N    :     J/  :  :    P^Q    :     P. 

For,  from  the  first  proportion,   we  have 

Mx  Q^XX  P,  or  .VX  P-J/X  ft 

Add  each  of  the  members  of  the  last  equation  to,  and 
subtract  it  from  JI X  P,   and  we  shall  have, 

MxP^XxP  =  MxP^MxQ;   or 

{M±:X)XP-={P^Q)XJf. 

But  J/=tiV  and  P,  may  be  considered  the  two  extremes, 
and  P^Q  and  J/,  the  two  means  of  a  proportion  (p.  2): 
hence, 

(J/dbX)     :     J/    :  :     (P±(?)     :     P. 

PROPOSITION  vii.    theorp:m. 

Jilquimultij^les  of  any  tiro   wag nifntlf^.^^  Jittve    the   same  ratio  as 
the  inag.iitadts  thtnisrlces. 

Let  Jf  and  A^  be  any  two  magnitudes,  and  m  any  num- 
ber Avhatever ;    then  will  mxM^  and  niXN^   be  equal  mid- 


5-i  GEOMETRY. 

tiples  of  M  and  N  \  then  mxM  will  be   to  mXN'j  in  the 
ratio  of  IT  to  iV! 

For,  MxN^NxMx 

multiplying  eacli  member  bj''  m,  and  we  have 

m  X  Mx  N=  m  X  N  X  M :    then  (p.  2), 
mXM  :    mXN  ::    M  :   K 

PROrOSlTlON  VIII.    TIIEOKEM. 

Of  four  proportional  magnitudes,  if  there  he  tclhen  any  eqimnul' 
tiples  of  the  two  anti-cedents^  and  any  equimultij)ks  of  the  tv)0 
consequents,  such  equimultiples  will  he  2^^'02)ortionaL 

Let  JTj  N',  P,  Q,  be  four  magnitudes  in  proportion ;  and 
let  m  and  n  be  any  numbers  whatever,  then  will 
mxJf    :     nX  X    :  :     mxP    :     nX  Q. 

For,  since         M    :     N    :  :     P    :     ft 
we  have  M xQ=XxP; 

hence,  m  X  Mx  n  X  Q=n  X  Xx  m  X  P, 

by  multiplying   both    members  of  the    equation    by  mXn. 
But  mxM  and  n  X  Q,  may  be  regarded  as  the  two  extremes, 
and  nXN  and  m  xP,  as  the  means  of  a  proportion  ;  hence, 
m  XM    :     nxX    :  :     m  xP    :     nXQ, 

PKOPOSITION    IX.     TIIEOKEM. 

Of  four  proportional  viar/nifudes,  if  the  two  consequents  he 
either  augmented  or  diini)ii-shed  hy  magnitudes  wliich  have 
the  same  ratio  as  the  ant/^cedents,  the  resulting  magnitudes 
and  tlie  antecedents  icill  he  proportional. 


Ixit 

M    :     N    : 

:     P    :     ft 

and  let 

M    :     P 

:     m     :     ri] 

then  will 

M    :     P 

:     iVzbm     :     Q±n. 

For,  since 

M    :     N 

:  :     P    :     ft  MxQ=XxP. 

and  since 

M    :     P 

:  :     m     :      n,  Mx  7i=PXm, 

therefore. 

Mx  Q^}fX  n  =  XxP±Px  m, 

or 

J\fX{Q±n)=Px(X^m): 

hence  (p.  2), 

M    :     P 

:  :     Xdcm     :     Q^n, 

BOOK    II 


I'KOPOSITION  X.     THEOREM. 


If  any  numher  cf  magnitudes  are  proportionals^  any  one  ante' 
cedent  vAll  he  to  its  consequent,  as  the  sum  of  all  the  ante- 
cedents  to  the  sum  of  the  consequents. 

Let      31    :     N-    ::    P    :     Q    ::    li    :     S,   kc. 

Then  since, 

3£  :  JSr  ::  P  :  Q,  WQ  have  J/x  §=.VxP, 
and,  31  :  A^  :  :  R  :  jS,  we  have  3IxS=^NxB, 
add  to  each  3fxN=3Ixy, 

then,     3Ix]Sr+3IxQ+3fxS=3IxN+jSrxP+NxP, 
or,  3Ix{.y+Q+S)=Nx{3f+P+B); 

therefore  (p.  2),  J/    :     N    :  :    3I+P+E    :     N+Q+S. 

PEOPOSITION   XI.     THEOREM. 

If  two  magnitudes  he  each  increased  or  diminished  hy  like 
parts  of  each,  the  resulting  magnitudes  will  have  the  same 
ratio  as  the  magnitudes  themselves. 

Let  3f  and  iV  be   any  two   magnitudes  and  —  and  — 

like  parts  of  each. 

We  have  J/xi\^=i/x7V^ 

....              .      ^^X^    ^^XiV  ,       ,  , 

add  to  both,  or  siibt.  = >  member  by  member 

'  m  m  '^ 

J/xiV      _     ,^,  3fxy 
and  we  have  (a.  2),      J/XxV±  —^  =31 XN^  ~^^' 


\         m/         \         m  / 


31  y 

that  is  (p.  2),  J/    :     JSr    ::    3I±—     :    N±—' 


PROPOSITION  XII.    THEOREM. 

If  four  magnitudes  are  proportional,  their  squares  or  cubes  wiU 
.    also  he  proportional. 


Let  31    '.     N    '.     P    \     Q, 

Then  will,  3IxQ=NxP. 


50  GEOMETRY. 

By  squaring  both  members,    3J''xQ'=N'xP  ^ 

and  by  cubing  both  members,  3J'  X  Q  =xV"  X  P'  ; 

therefore,  J/"     :     N^     :  :     P"     :      (>", 

lui  2f    :     .V'     ::     P'     :     §'• 

O'/-.    In  a  similar  v/ay  it  may  be  shown  tliat  like  pON^era 
or  roots  of  proportional  magnitudes  are  proportionals. 

PKOPUSITIUN  XIII.     THEOREM. 

If  tliei^  he  two  sets  of  jyroportlonal  marjnitudes,  the  products  of 
the  correspond  in  rj  terms  will  be  proportionals. 

Let  M    :     F    ::     P    :     Q, 

and  P    :      S    ::      T    :      V, 

then  AviU        21 XP    :     XxS    ::     PxT    :     QxV. 

For,  since  JJxQ=XxP, 

and  PxV=^Sx7] 

we  shall  have  31 X  QxPx  V  =  Xx  PxSxT, 


MxRxQx  V^XxSxPxT] 


therefore,        MxP    :     ^Vx^'    ::     PxT    :      QxV. 


PROPOSITION    XIV.     THEOREM. 

If  any  vn.mher  of  magnitudes  are  continued  jnvjwrtionals  ;  then, 
the  ratio  of  the  first  to  tJie  third  ivill  he  duplicate  of  the 
common  ratio  ;  and  the  ratio  of  die  first  to  the  fourdi  wili 
he  triplicate  of  the   common  ratio ;   and  so  on. 

For    let  A  be  the  first  term,  and  m  the  common  ratio: 
the  proportional  magnitudes  will  then  be  represented  by 

A^  m^xA,  nrxA,  m'^xA^  m'^xA^  &c. : 

Kow,  the  ratio  of  the  first  to  any  one  of  the  following 
terms  exactly  corresponds  with  the  eriiu cation. 


BOOK    III 


THE  CIRCLE,  AND  THE  MEASUREMENT  OF  A.NGLES. 


DEFINITIONS, 


1.  Tlie  Circumference  of  a  Circle 
is  a  curve  line,  all  the  points  of  which 
are  equally  distant  from  a  point  within, 
called  the  centre. 

The  circle  is  the  portion  of  the  plane 
terminated  by  the  circumference. 

2.  Every  straight  line,  drawn  from  the  centre  to  the 
circumference,  is  called  a  nalut.s,  or,  seinidiaiiifUn\  Every 
line  which  passes  through  the  centre,  and  is  terminated,  od 
both  sides,  by  the  circumference,  is  called  a  dia meter. 

From  the  definition  of  a  circle,  it  follows,  that  all  the 
radii  are  equal ;  that  all  the  diameters  are  also  e(iual,  and 
each  double  the  radius. 

3.  Any  part  of  the  circumference  is  called  an  arc.  A 
straight  line  joining  the  extremities  of  an  arc,  and  not  passing 
through  the  centre,  is  called  a  Jiurd,  or  suUeiuse  of  the  arc* 

4.  A  Segment  is  the  part  of  a  circle  included  between 
an  arc  and  its  chord. 

5.  A  Sector  is  the  part  of  the  circle  included  between 
an  arc,  and  the  two  radii  draw^n  to  the  extremities  of  the 
arc. 


*  In  all  cases,  the  same  chord  belongs  to  two  arcs,  mid  c  nisccjiieutiy,  also  to  twc 
ae^rmeuts  :   but  the  smaller  one  is  aJwajs  meant,  nnless  the  o  n'-rury  ^^*  «xi  rossed. 


5S 


GEOMETEY. 


6.  A  Straiciit  Line  is  said  to  be 
inscrlhtd  in  a  circle^  when  its  extremities 
are  in  the  circumference. 

An  inscrihed  a:\fjle  is  one  Avliicli  has 
its  vertex  in  the  circumference,  and  is 
mcluded  bv  two  chords  of  the  circle. 


7.  An  inscrihed  triangle  is  one  which 
has  the  vertices  of  its  three  angles  in  the 
circumt'erence. 

And  generally,  a  polygon  is  said  to 
be  viscriUd  in  a  circle,  when  the  vertices 
of  all  its  angles  are  in  the  circumfer- 
ence. The  circumference  of  the  circle 
is  then  said  to  circumscribe  the  polygon. 

8.  A  Secant  is  a  line  which  meets 
the  circumference  in  two  points,  and  lies 
partly  within,  and  parti}'  without  the  cii'cle. 

9.  A   Taxgent   is  'a    line    which  has 

but  one   point   in    comu:ion   Avith  the    cir-      ^ 

cumference. 

The  })()int  where  the  tangent  touches  the  circumference, 
is  called  the  j^^^'d  of  contact. 

10.  Two  circumferences  tonch  each 
other  when  they  have  but  one  point  in 
common.  The  common  point  is  called 
the  point  of   tangtncg. 


11.  A  polygon  is  circumscribed  about  a 
circle,  when  each  of  its  sides  is  tangent  to 
the  circumference.  l:i  the  same  case,  the 
circle  is  said  to  be  inscribed  in  the  poly- 
gon. 


POSTULATE. 

12.  Let  it  be  granted  that  the  circumference  of  a  circle 
may  be  described  from  any  centre,  and  with  any  radius. 


BOOK    III, 


59 


PEOrOSITION  I.     THEOREM. 

Every  diaraeter  divides  the  circle  and  its  circumference  each  into 
two  equal  2'>ci^ts. 

Let  AEBF  be  a  circle,  and  AB  a,  diameter.  ISTow,  if 
the  figure  AEB  be  applied  to  AFBj 
their  common  base  AB  retaining  its 
position,  the  curve  line  AEB  must 
fall  exactly  on  the  curve  line  AFB^ 
otherwise  there  would,  in  the  one  or 
the  other,  be  points  unequally  dis- 
tant from  the  centre,  which  is  con- 
trary to  the  definition  of  a  circle.  Ilence,  the  diameter 
divides  the  circle  and  its  circumference,  each  into  two 
equal  parts. 


PKOPOSITIOX  II.      THEOREM. 

Every  chord,  is   less   than   a  diameter. 

Let  AD  \)Q  any  chord.     Draw  the 
radii  C/1,  Cl)^  to  its  extremities.     We 
shall  then  have  (b.  i.,  p.  7)* 
AD<AC-\-CD, 

but  AC  plus  CD   is   equal   to   AB] 
hence,  AD<^AB. 

Cor.    Ilence,  the  greatest  line  which  can  be  inscribed  in 
a  circle  is  a  diameter. 


PROPOSITION   III.     THEOREM. 

A  straight    line   cannot   meet    the   circumference   of  a    circle    in 
more  than  tivo  points. 

For,  if  it  could  meet  it  in  three,  those  three  points 
would  be  equally  distant  from  the  centre;  and  there  would 
be  three  equal  straight  lines  di-awn  from  the  same  point  to 
the  same  straight  line,  which  is  impossible  (b.  i.,  p.  15,  c.  2). 

.  *  When  reference  is  made  from  one  Proposition  to  tnotlier,  in  tlie  .*/w/'  Boole,  tlie 
number  of  tlie  Proposition  referred  to  Is  alone  jriven;  out  wlicu  tlie  Proposition  i» 
found  Ln  a  ditt'ercut  Book,  the  number  of  the  Booli  is  also  given. 


60 


GEOMETRY. 


PROrOSITION    IV.     TIIEOKEil. 

Tn  the  fiame  cirdc^  or  in  eqnal  circles,  equal  arcs  are  snht^-nded  by 
equal  chords :  and  corwersehj^  ^q^ud  chords  suhicnd  equnl  vrcs. 
Let  C  and   0  be    the   centres  of  two  equal  circles,  and 

suppose    tiie    arc   AMD  equal  to    the    arc  FXG  :    then  will 

the  chord  A/J  be  efjual  to  the  chord  £G.  - 
For,  since  the  diam- 


eters  AB,  A7%  are  equal, 

the    se m  i  -ci  rcl e    A  MDB 

may  be   applied    to    the 

semi-circle    EXGF\    and 

the    curve    line    A  MDB 

"will    coincide    with    the 

curve  line  EXGF.     But  the  part  AMD  is  equal  to  the  part 

EXG^  by  hypothesis ;    hen<^e,  the   point  D  will   fall  on  G  ; 

therefore,  the  chord  AD  will  coincide  with  EG  (b.  l,  a.  11), 

and  hence,  is  equal  to  it  (b.  i.,  a.  14). 

Coiivei'sel//:  If  the  chord  AD  is  equal  to  the  chord  EG, 
the  subtended  arcs  AMD,  EXG,  will  also  be  equal. 

For,  drawing  the  radii  CD,  OG,  the  triangles  ACD,  EGG, 
will  have  their  sides  equal,  each  to  each,  namely,  AC=EO, 
CD=OG,  and  AD=EG;  hence,  the  triangles  are  them- 
selves equal  ;  and,  consequently,  the  angle  A  CI)  is  equal 
to  EGG  (b.  L,  p.  10.) 

Xow,  place  the  semi-circle  ADD  on  its  equal  EGF,  so 
that  the  radius  AC  may  fall  on  the  equal  radius  EG. 
Then,  since  the  angle  ACD  is  equal  to  the  angle  EGG,  the 
radius  CD  will  faU  on  GG,  and  the  sector  AMDC  will 
coincide  with  the  sector  EXGG,  and  the  arc  AMD  with 
the  arc  EXG  :  therefore,  the  arc  AMD,  is  equal  to  the  arc 
EXG  (B.  I.,  A.  14). 


rROPOSITTON  V.     TIIEOEEM. 

In  eoual  circles,  or  in  the  same  circle,  a  gy eater  arc  is  subtend' 
nd   by   a   greater   chord:   and   conversely,   the  greater    chord 
subtends  the  grecUer  arc. 
Let  C  be  the  common  centre  of  two  equal  circles :  then, 

if  the  arc  ADH  is  greater  than  the  arc  AD,  the  chord  AH 

wiU  be  greater  than  the  chord  AD. 


BOOK   III. 


61 


For,  draw  the  radii  CA,  CD, 
CH,  and  tlie  chords  AD,  AIL 
Now,  the  two  sides  AC,  ClI,  of 
the  triangle  AC II  are  equal  to 
the  two  sides  AC,  CD,  of  the  tri- 
angle ACD,  and  the  angle  ACII, 
is  greater  than  ACD'.  hence,  the 
third  side  All  is  greater  than  the 
third  side  AD  (b.  I.,  P.  9);  there 
fore  the  chord  which  subtends  the  greater  arc  is  the  greater. 

Conversely:  If  the  chord  AH  \^  greater  than  AD,  the 
arc  AD II  will  be  greater  than  the  arc  AD. 

For,  if  ADII  were  equal  to  AD,  the  chord  All  would 
be  equal  to  the  chord  AD  (p.  4),  which  is  contrary  to  the 
h3^pothesis :  and  if  the  arc  ADII  were  less  than  AD,  the 
chord  All  would  be  less  than  AD,  which  is  also  contrary 
to  the  hypothesis.  Then,  since  the  arc  ADll,  subtended  by 
the  greater  chord,  cannot  be  equal  to,  nor  less  than  AD,  it 
must  be  greater. 

Scholium.  The  arcs  here  treated  of  are  each  less  than 
the  semi-circumference.  If  tJiey  were  greater,  the  reverse 
property  would  have  place ;  for,  as  the  arcs  increase,  the 
chords  will  diminish,  and  conversely. 


FKOPOSITION   VI.     THEOREM. 


The  radius  which  is  perpendicular  to  a  chord,  bisects  the  chord^ 
and  bisects  also  the  subtended  arc  of  the  chord. 

Let  AB  be  any  chord,  and  GQ  a  radius  perpendicu- 
lar to  it:  then  will  AD  be  equal  to  DB,  and  the  arc  AQ 
to  the  arc  (IB. 

For,  draw  the  radii  CA,  CB. 
Then  the  two  right-angled  trian- 
gles ADC,  CDB,  will  have  AC 
equal  to  GB,  and  CD  common ; 
hence,  AB  is  equal  to  DB  (b.  i., 
P.  17). 

Again,  since  AD,  DB,  are 
equal,     GO-     is     a    perpendicular 


GEOMETRY. 


erected  from  the  middle  of  AB; 

and  since  6^   is   a   point  of  this 

perpendicular,    the     chords    AG 

and  GB   are   eqnal  (b.  i.,  p.  16). 

But    if  the    chord  AG   is  equal 

to    the    chord  GB,  the   arc   AG 

is   equal   to  the  arc    GB   (p.  4) ; 

hence,   the  radius    CG,    at    right 

angles  to  the  chord  AB,  divides 

the   arc   subtended   by   that    chord  into  two  equal  parts. 

Scholium.  The  centre  G,  the  middle  point  I)  of  the 
chord  AB,  and  the  middle  point  G  of  the  subtended  arc, 
are  three  points  of  the  same  straight  line  perpendicular  to 
the  chord.  But  two  points  determine  the  position  of  a 
straight  line  (a.  11) ;  hence,  every  straight  line  which  passes 
through  two  of  these  points,  will  necessarily  pass  through 
the  third,  and  be  perpendicular  to  the  chord. 

It  follows,  also,  that  tJie  perpendicular  raised  at  the  middle 
point  of  a  chord  passes  through  the  centre  of  the  circle,  and 
through  tJie  middle  point  of  the  suhtended  arc. 

For,  the  perpendicular  to  the  chord,  drawn  from  the 
centre  of  the  circle,  passes  through  the  middle  point  of  the 
chord,  and  only  one  perpendicular  can  be  drawn  from  the 
same  point  to  the  same  straight  line  (b.  I.,  P.  14,  c). 


PEOPOSITION  VII.     THEOEEM. 

Througa   three   given  points,  not  in  the  same  straight  line,  one 
circumference  may  cdicays  he  made  to  pass,  and  hut  one. 

Let  A,  B,  and  C,  be  the  given  points. 

Join  the  points  A  and  B  by 
the  straight  line  AB,  and  the 
points  B  and  G  by  the  straight 
line  BO,  and  then  bisect  these 
lines  by  the  perpendiculars  DE 
FG\  we  say  first,  that  DE  and 
FG,  will  intersect  in  some  point  0. 

For,  they  intersect  each  other 
unless   they   are   parallel   (b.  l,  d.  16).     Now,    if  they   are 


BOOK    III.  63 

parallel,  the  line  AB  whicli  is  perpendicular  to  BE,  is  also 
perpendicular  to  FG,  and  the  angle  ^  is  a  right  angle 
(b.  I.,  p.  20,  c.  1).  But  BK,  the  prolongation  of  AB,  is  a 
different  line  from  BF,  because  the  three  points  A,  B,  C, 
are  not  in  the  same  straight  line ;  hence,  there  would  be 
two  perpendiculars,  BF,  BK,  let  fall  from  the  same  point 
B,  on  the  same  straight  line,  which  is  impossible  (b.  I.,  P. 
14) ;  hence,  DF,  FQ,  are  not  parallel,  and  consequently, 
will  intersect  in  some  point  0. 

Moreover,  since  the  point  0  lies  in  the  perpendicular  DE^ 
it  is  equally  distant  from  the  two  points,  A  and  B  (b.  i., 
p.  16);  and  since  the  same  point  0  lies  in  the  perpendicu- 
lar FG,  it  is  also  equally  distant  from  the  two  points  B 
and  C:  hence,  the  three  distances  OA,  OB,  00,  are  equal; 
therefore,  the  circumference  described  from  the  centre  0, 
with  the  radius  OB,  will  pass  through  the  three  given 
points,  A,  B,   C. 

Wc  have  now  shown  that  one  circumference  can  always 
be  iiade  to  pass  through  three  given  points,  not  in  the 
same  straight  line  :  we  say  fiirther,  that  but  one  can  be 
described  through  them. 

For,  if  there  were  a  second  circumference  passing  through 
the  three  given  points  A,  B,  0,  its  centre  could  not  be  out 
of  the  line  DE,  for  any  point  out  of  this  lino  is  unequally 
distant  from  A  and  B  (b.  I.,  P.  16) ;  neither  could  it  be  out 
of  the  line  FG,  for  a  like  reason ;  therefore,  it  would  be 
in  both  the  lines  DE,  FG.  But  two  straight  lines  cannot 
cut  each  other  in  more  than  one  point ;  hence,  there  is  but 
one  circumference  which  can  pass  through  three  given 
points. 

Cor.  Two  circumferences  cannot  meet  in  more  than  two 
points ;  for,  if  they  have  three  common  points^  thero  wdll 
be  two  circumferences  passing  through  the  s.uno  vhreo 
points ;  which  has  been  shown,  by  the  propos.'vcTi.  tt>  be 
impost'ibie. 


64 


geomp:try 


i'KOrosiTloX  VIII.     TIIEOKE^L 

Two  equal  dionls  are  eqanlh/  distant  from  the  centre ;  and  of 
two  'i/ie/jaal  ckurds^  tlit  less  is  at  the  fjreater  distance  from 
the  centre. 

Suppose  the  chord  AB  to  be  equal  to  the  chord  DE. 
From  C  the  centre  of  the  circle,  di-aw  OF,  and  CG  res- 
pectively ])erpendicular  to  the  chords :  then  will  CF  be 
3qual  to  C(jr. 

Draw  the  radii  CA^  CD ;  then 
'n  the  right-angled  triangles  CAF^ 
DOG,  the  hypothenuses  CA,  CD^ 
are  equal  (d.  2) ;  and  the  side 
AF,  the  half  of  AB  (p.  6),  is 
equal  to  the  side  DG^  the  half 
of  DE:  hence,  the  triangles  are 
equal,  and  CF  is  equal  to  CG 
(b.  I.,  p.  17) ;  concequently,  the 
two  equal  chords  AB^  DE^  are  equally  distant  from  the 
centre. 

Secondhj.  Let  the  chord  AH  be  greater  than  DE:  then 
will  DE  be  furthest  from  the  centre  C.  Since  the  chord 
AH  is  greater  than  DE  ths  arc  AKII  is,  greater  than  DME 
(p.  5).  Cut  off  from  the  former,  a  part  AXB^  equal  to 
DME;  draw  the  chord  AB,  and  draw  CF  perpendicular 
to  this  chord,  and  CI  perpendicular  to  AIL  It  is  evident 
that  CF  is  greater  than  CO  (b.  I.,  A.  8),  and  CO  than  CI 
(b.  I.,  P.  15)  ;  therefore,  CF  is  still  greater  than  CI.  But 
CF  is  equal  to  CG^  because  the  chords  AB,  DE,  are  equal : 
hence,  CG  is  greater  t<lian  CI;  therefore,  of  two  unecjual 
chords,  the  less  is  the  farther  from  the  centre  of  ihe  circle. 


PEOPosrnoN  ix.    theokem. 

A  straight   line  perpendicular  to  a  radius,    at  its    ixtijmity,  is 
tangent  to  the  circumference. 

Let  the  line  BD  be  perpendicular  to  the  radius  CA  at 
its  extremity  A  ;  then  will  it  be  tangent  to  the  circumfer- 
ence. 


BOOK    III 


65 


For,  every  oblique  line  CE^  P> 
is  longer  tlian  the  perpendicular 
CA  (b.  I.,  p.  15) ;  hence,  the  point 
E  is  without  the  circle;  there- 
fore, the  line  BD  has  no  point 
but  A  in  common  with  the  cir- 
cumference;   consequently,  the  line  BD  is  a  tangent  (d.  9). 

Cor.  1.  Conversely,  if  a  straight  line  be  tangent  to  a 
circle,  it  will  be  perpendicular  to  the  radius  drawn  to 
the  point  of  contact. 

Let  BAD  be  a  tangent,  and  CA  a  radius  drawn 
through,  the  point  of  contact  A :  then  mil  BD  be  perpen- 
dicular to  CA.  For,  through  the  centre  (7,  suppose  any 
other  line,  as  COE^  to  be  drawn.  Then,  since  BD  is  a 
tangent,  the  point  E  will  lie  without  the  circle,  and  conse- 
quently CE  will  be  greater  than  the  radius  CO  or  CA ; 
therefore,  the  radius  CA,  measures  the  shortest  distance 
from  the  centre  C,  to  the  tangent  BD:  hence,  it  is  per- 
pendicular to  the  tangent  (b.  l,  p.  15,  c.  1). 

Cor.  2.  At  a  given  point  of  the  circumference  only  one 
tangent  can  be  drawn  to  the  circle.  For,  let  A  be  the 
given  point,  BD  a  tangent,  and  CA  the  radius  drawn 
through  the  point  of  contact  A.  N'ow,  if  another  tangent 
could  be  drawn,  it  would  also  be  perpendicular  to  CA  at 
the  point  A,  by  the  last  corollary:  that  is,  we  should  have 
two  lines  perpendicular  to  CA,  at  the  same  point;  which 
is  impossible  (b.  i.,  p.  14,  c). 


PROPOSITION  X.     THEOEEM. 
Two  2^<^(''^(^^LleIs  intercept  equal  arcs  of  the  circumfei 

There  may  be  three  cases. 

First.  When  the  two  parallels 
are  secants.  Let  AB  and  DE  be 
two  parallels :  draw  the  radius  CH 
perpendicular  to  the  chord  MP. 
It  will,  at  the  same  time,  oe  per- 
pendicular to  NQ  (b.  I.,  p.  20,  c.  1) ; 
therefore,  the    point  H  will  be  at 


•ence. 


(56 


GEOMETRY. 


once  the  middle  of  tlie  arc  MHP, 
and  of  tlie  arc  NIIQ  (p.  6) ;  conse- 
quently, we  shall  have  the  arc 
2fH=IIP^  and  the  arc  NH=HQ', 
rind  therefore 

MH-NH=HP-HQ ; 
Ji  ©ther  words,  MN=PQ. 

Second.  When,  of  the  two  par- 
allels AB^  DE^  one  is  a  secant,  and 
the  other  a  tangent,  draw  the  radius 
CH  to  the  poin1>  of  contact  H]  it 
will  be  perpendicular  to  the  tan- 
gent DE  (p.  9,  c.  1),  and  also  to 
its  parallel  MP  (b.  l,  p.  20,  c.  1). 
But  since  CH  is  joerpendicular  to 
the  chord  J/P,  the  point  H  must 
be  the  middle  of  the  arc  MHP  (p.  6)  ;  therefore,  the  arcs 
MH^  HP^  included  between  the  parallels  AB^  JDE^  are 
equal. 

Third.  If  the  two  parallels  DE^  IL^  are  tangents,  the 
one  at  H^  the  other  at  K^  draw  the  parallel  secant  AB ; 
and,  from  what  has  just  been  shown,  we  shall  have 

2IH=HP,  MK=KP: 
and  hence,  the  whole  are  HMK=HPK.     It  is  further  evi- 
dent that  each  of  these  arcs  is  a  semi-circumference. 

Cor,  Conversely :  If  the  arc  HM  is  equal  to  the  arc 
77P,  it  is  plain  that  the  chord  MP  will  be  parallel  to  the 
tangent  DE. 


PKOPOSITION  XI.     THEOEEM. 

If  two  circumferences  have  one  jioint  common,  out  of  the  straight 
lirie  which  joins  their  centres,  they  will  also  have  a  secorJi 
point  in  common  ;  and  the  two  points  ivill  he  situated  in  a 
line  perpendicular  to  the  line  joining  the  centres^  and  ai 
equal  distances  from  it. 

Let  the  two   circumferences  described  about  the  centres 
0  and  JD  intersect    each    other   at  the  point  A ;   draw  AF 


BOOK    III, 


67 


perpendicular  to  CD^  and  prolong  it  till  BF  is  equal  to 
AF  \  then  will  the  circumferences  also  intersect  each  other 
at  B. 


For,  since  AF  is  equal  to  FB^  CF  common  and  the 
angles  at  F  right  angles,  the  hypothennses  CB  and  GA  are 
equal  (b.  I.,  p.  5) :  hence,  the  circumference  described  about 
the  centre  G^  with  the  radius  GA^  will  pass  through  B. 
In  the  same  manner  it  may  be  shown,  that  the  circumfer- 
ence described  about  the  centre  Z^,  with  the  radius  DA^ 
will  also  pass  through  B. 

Gov.  If  two  circumferences  intersect  each  other,  they 
will  intersect  in  two  points,  and  the  line  which  joins  the 
centres  will  be  perpendicular  to  the  common  chord  at  the 
middle  point. 


PKOPOSITION  XII.    THEOREM. 

If  the  circumferences  of  two  circles  intersect  each  other ^  the  dis- 
tance between  their  centres  will  he  less  than  the  sum  of  their 
radii^  and  greater  than  the  difference. 


Let  two  circumferences  be 
described  about  the  centres  G 
and  Z^,  with  the  radii  GA  and 
DA :  then,  if  these  circumfer- 
ences intersect  each  other,  the 
triangle  GAD  can  always  be 
formed.  Now,  in  this  triangle, 
GAD, 


also, 


GD<GA-{-AD  (b.  L,  p.  7), 
GD>DA-AG  {b,i.,  p.  7,  c.) 


6S 


GEOMETRY. 


PBOPOSITIOX  Xm.     THEOREiL 


Ij  Hie  distance  between  the  centres  of  two  circles  is  equal  to 
the  sum  of  their  radii,  the  circumferences  v:ill  touch  each 
other  externally. 

Let  C  and  D  be  the  centres  of  two  circles  at  a  distance 
from  each  other  equal  to  CA-\-AD. 

The  circles  will  evidently  have 
the  point  A  common,  and  thev  will 
have  no  other  ;  because  if  they  have 
two  points  conmion,  the  distance 
between  their  centres  must  be  less 
than  the  sum  of  their  radii,  which 
is  contrary  to  the  supposition. 

Cor.  K  the  distance  between  the  centres  of  two  circles 
is  greater  than  the  sum  of  their  radii,  the  two  circumfer- 
ences will  be  exterior  the  one  to  the  other. 


PKOPOSITION   XR'.     THEOEEM. 


If  the  distance  hetween  the  centres  of  two  circles  is  equal  to  the 
difference  of  their  radii,  the  two  circumferences  will  toucJi 
each  other  internally. 

Let  C  and  D  be  the  centres  of  two  circles  at  a  distance 
from  each  other  equal  to  AD—  CA. 

It  is  evident,  as  before,  that  the 
two  circumferences  will  have  the 
point  A  common :  they  can  have 
no  other ;  because  if  they  had,  the 
distance  between  the  centres  would 
be  greater  than  AD—  CA  (p.  12)  ; 
which  is  contrary  to  the  supposition. 

Cor.  1.  Hence,  if  two  circles  touch  each  other,  either 
externally  or  internally,  their  centres  and  the  point  of  con- 
tact v.'ill  be  in  the  same  straight  line. 

Ccr.  2.    K   the    distance   between    the    centres   of   two 


BOOK    III.  69 

circles  is  less   tlian   the  difference  of  tlielr  radii,  one  circle 
will  be  entirely  witliin  the  other. 

Scholium  1.  All  circles  which  have  their  centres  on  the 
right  line  AD^  and  which  pass  through  the  point  A^  are 
tangent  to  each  other  at  the  point  A.  For,  they  have  only 
he  point  A  common,  and  if  through  A^  AE  be  drawn 
perpendicular  to  AD^  it  will  be  a  common  tangent  to  all 
the  circles. 

Scholmnn.  2.  Two  circumferences  must  occupy  with  res- 
pect to  each  other,  one  of  the  five  positions  above  indi- 
cated. 

1st.  They  may  intersect  each  other  in  two  points  : 
2d    They  may  touch  each  other  externally: 
3d   They  may  be  external,  the  one  to  the  other : 
AitJi.  They  may  touch  each  other  internally  : 
bth.  The  one  may  be  entirely  within  the  other. 


PKOPOSITION   XV.     THEOREM. 

In  the  same  circle^  or  in  equal  circles,  radii  making  equal 
angles  at  Hie  centre,  intercept  equal  arcs  on  the  circumference. 
And  conversely :  If  the  arcs  intercepted  are  equal,  the  angles 
contained  hy  the  radii  are  also  equal. 

Let  G  and  C  be  the  centres  of  equal    circles,  and  the 
angle  AQB^DCE. 

First.  Since  the  angles 
ACB,  DOE,  are  equal,  one  of 
them  may  be  placed  upon 
the  other.  Let  the  angle  A  CB 
be  placed  on  DOE.  Then 
since  their  sides  are  equal, 
the  point  A  will  evidently  fill  on  D,  and  the  point  B  on 
E.  The  arc  AB  will  also  fall  on  the  arc  BE;  for,  if  the 
arcs  did  not  exactly  coincide,  there  would,  in  the  one  ot 
the  other,  be  points  unequally  distant  from  the  centre; 
which   is    impossible :    hence,  the    arc   AB   is  equal  to  DE 

(A.  14). 

jStcond.   K  the   arc   AB=DE,  the    angle   ACB    is   equal 


70 


GEOMETRY, 


to  DCE.  For,  if  tliese  angles  are  not  equal,  suppose 
one  of  them,  as  ACB^  to  be  the  greater,  and  let  ACI  be 
taken  equal  to  DCE.  From  what  has  just  been  shown, 
we  shall  then  have  AI=  DE  \  but,  by  hypothesis,  AB  is 
equal  to  DE  ]  hence,  AI  must  be  equal  to  AB^  or  a  part 
equal  to  the  whole,  which  is  absurd  (a.  8) ;  hence,  the 
angle  ACB  \&  equal  to  DCE. 


PEOPOSITION   X^^.     TUtlUKEM. 

In  the  same  circle^  or  in  equal  circles^  if  two  angles  at  the 
centre  have  to  each  other  the  ratio  of  two  ichole  numbers^ 
the  intercepted  arcs  will  have  to  each  other  the  same  ratio: 
or,  ice  shall  have  the  angle  to  the  angle^  as  the  corresjpond- 
ing  arc  to  the  corresponding  arc. 

Suppose,  for  example,  that  the  angles  ACB^  DCE,  are 
to  each  other  as  7  is  to  4 ;  or,  what  is  the  same  thing, 
suppose  that  the  angle  JiJ  which  may  serve  as  a  common 
measure,  is    contained  7  times   in   the   angle  ACB^    and  4 

G 
M 


times  in  DCE.  The  seven  partial  angles  A  Cm,  mCn,  nCp^ 
&c.,  into  which  ACB  is  divided,  are  each  equal  to  any  of 
the  four  partial  angles  into  which  DCE  is  di^dded ;  and 
each  of  the  partial  arcs.  Am,  mn,  np,  &;c.,  is  equal  to  each 
of  the  partial  arcs  Dx,  xy,  &;c.  (p.  15).  Therefore,  the  whole 
arc  AB  will  be  to  the  whole  arc  DE,  as  7  is  to  4.  But 
the  same  reasoning  would  evidently  apply,  if  in  place  of 
7  and  4  any  numbers  whatever  were  employed  ;  hence,  if 
the  angles  ACB,  DCE,  are  to  each  other  as  two  whole 
numbers,  they  will  also  be  to  each  other  as  the  arcs  AB^ 
DE. 

Cor.   Conversely :  If  the  arcs  AB,  DE,  are  to  each  othei 
as   two   whole   numbers,  the  angles  ACB^  DCE  will  be  tc 


BOOK    III.  71 

ea.ch  other  as  tlie  same  whole  numbers,  and  we  shall  ha^ve 

AB    :     DE    ::     ACB    :     DCE. 
For,  the   paitial  arcs,  Am^  mn^  &c.,  and  Dx^  xy^  kc,  being 
equal,  the  partial  angles  AOm^  mCn^  &c.,  and  DCx,  xCy^  &;c., 
will  also  be   equal,   and   the   entire    arcs    will   be   to    eacli 
other    as  the  entire  angles. 

PEOrOSITION  XVII.      THEOEEM. 

In  the   same   circle^  or   in    equal  circles^  any  two   angles  at  tlie 
centre  are  to  each  oilier  as  tlie  intercepted  arcs. 

Let   ACB  and  ACD   be   two    angles   at   the  centres  of 
equal  circles :    then  will 

ACB    :     ACB 

For,  if  the  angles  are 
equal,  the  arcs  will  be  equal 
(p.  15):  If  the  J  are  unequal, 
let  the  less  be  placed  on 
the  greater.  Then,  if  the 
proposition  is  not  true,  the 
angle  A  CB  will  be  to  the  angle  A  CD  as  the  arc  AB  is  to  an 
arc  greater  or  less  than  AD.  Suppose  such  arc  to  be 
greater,  and  let  it  be  represented  \>j  A0\  we  shall  thus 
have, 
the  angle  ACB  :  angle  ACD  :  :  arc  AB  :  arc  AO. 
Next  conceive  the  arc  AB  to  be  divided  into  equal  parts,  each 
of  which  is  less  than  DO;  there  will  be  at  least  one  poin? 
of  division  between  D  and  0 ;  let  /  be  that  point ;  and 
draw  CI.  Then  the  arcs  AB^  AI^  will  be  to  each  other 
as  two  whole  numbers,  and  by  the  preceding  theorem,  we 
shall  have, 

angle  ACB    :     angle  ACI    :  :     arc  AB    :     arc  AI. 
Comparing   the   two   proportions   with   each   other,  we   see 
that  the  antecedents  in  each  are  the  same :  hence,  the  cod- 
sequents  are  proportional  (b.  ii.,  p.  4) ;   and  thus  we  find, 

the  angle  ACD     :     angle  ACI    :  :     arc  AO     :     arc  AI. 
But  the  arc  AO  is  gTcater  than  the  arc  AI ;    hence,  if  this 
proportion  is  true,  the" angle  ACD  must  be  greater  than  the 


72  GEOMETEY. 

ungle  ACT:  on  tlip  contrarv,  however,  i:  is  less;  hence, 
the  aiagie  ACB  cannot  be  to  the  angle  ACD  as  the  arc  AB 
is  to  an  arc  greater  than  AD. 

By    a    process    of  reasoning   entirely  similar,  it  may  be 
shown   that    the   fourth   term  of  the    proportion  cannot  be 
ess  than  AD\   hence,  it  is  AD  itself;   therefore,  we  have 
angle  ACB    :     angle  ACD    :  :     arc  AB    :     arc  AD, 

Scholium  1.  Since  the  angle  at  the  centre  of  a  circle, 
and  the  arc  intercepted  by  its  sides,  have  such  a  connec- 
tion, that  if  the  one  be  augmented  or  diminished,  the  other 
will  be  augmented  or  diminished  in  the  same  ratio,  we  are 
authorized  to  assume  the  one  of  these  magnitudes  as  the 
measure  of  the  other ;  and  we  shall  henceforth  assume  the 
arc  AB  as  the  measure  of  the  angle  ACB.  It  is  only  neces- 
sary, in  the  comparison  of  angles  with  each  other,  that  the 
arcs  which  serve  to  measure  them,  be  described  with  equal 
radii. 

Scholiura  2.  An  angle  less  than  a  right  angle  will  be 
measured  by  an  arc  less  than  a  quarter  of  the  circumfer- 
ence :  a  right  angle,  by  a  quarter  of  the  circumference : 
and  an  obtuse  angle  by  an  arc  greater  than  a  quarter,  and 
less  than  half  the  circumference. 

Scholiura  3.  It  appears  most  natural  to  measm^e  a  quan- 
tity by  a  quantity  of  the  same  species ;  and  upon  this 
principle  it  would  be  convenient  to  refer  all  angles  to  the 
right  angle.  This  being  made  the  unit  of  measure,  an 
acute  angle  would  be  expressed  by  some  number  between 
0  and  1 ;  an  obtuse  angle  by  some  number  between  1  and  2. 
This  mode  of  expressing  angles  would  not,  however,  be 
the  most  convenient  in  practice.  It  has  been  found  more 
simple  to  measure  them  by  the  arcs  of  a  circle,  on  account 
of  the  facility  with  which  arcs  can  be  made  to  correspond 
to  angles,  and  for  various  other  reasons.  At  all  events,  if 
the  measurement  of  angles  by  the  arcs  of  a  circle  is  in 
any  degree  indirect,  it  is  still  very  easy  to  obtain  the  direct 
and  absolute 'measure  by  this  method;  since,  by  comparing 
the  fourth  part  of  the  circumference  wdth  the  arc  which 
serves  as  a  measure  of  any  angle,  we  find  the  ratio  of  a 
right  angle  to  the  given  angle,  which  is  the  absolute  measure. 


BOOK    III, 


78 


Sclwlium  4.  All  that  has  been  demonstrated  in  tne  last 
three  propositions,  concerning  the  comparison  of  angles 
with  arcs,  holds  true  equally,  if  applied  to  the  comparison 
of  sectors  with  arcs.  For,  sectors  are  not  only  equal  when 
their  angles  are  so,  but  are  in  all  respects  proportional  to 
heir  angles ;  hence,  two  sectors  A  OB^  A  GD^  taken  in  the 
same,  circle,  or  in  equal  circles,  are  to  each  other  as  the  arcs 
AB,  AI)j  the  bases  of  those  sectors.  Ilence,  it  is  evident  that 
the  arcs  of  equal  circles,  which  serve  as  a  measure  of  cor- 
responding angles,  are  proportional  to  their  sectors. 


PKOPOSITION  XVIII.     TIIEOEEM. 

Any    inscribed    angle    is    measured    by    half  the    arc    included 
between  its  sides. 

.  Let  BAD  be  an  inscribed  angle,  and  let  us  first  sup- 
pose the  centre  of  the  circle  to  lie  within  the  angle  BAD. 
Draw  the  diameter  AGE^  and  the  radii  GB^  GD. 

The  angle  BGE^  being  exterior  to 
the  triangle  ABG^  is  equal  to  the  sum 
of  the  two  interior  angles  GAB^  ABG 
(b.  I.,  P.  25,  c.  6) :  but  the  triangle  BA  G 
being  isosceles,  the  angle  GAB  is  equal 
to  ABG  ]  hence,  the  angle  BGE  is  double 
BAG.  Since  BGE  is  at  the  centre, 
it  is  measured  by  the  arc  BE  (p.  17,  s.  1); 
hence,  BAG  will  be  measured  by  the 
half  of  BE.  For  a  like  reason,  the  angle  GAD  will  be 
measured  by  the  half  of  ED]  hence,  BAG  +  GAD,  or  BAD 
will  be  measured  by  half  of  BE-\-ED,  or  lialf  of  BED. 

Secondly.  Suppose  the  centre  G  to 
lie  without  the  angle  BAD.  Then,  draw- 
ing the  diameter  AGE,  the  angle  BAE 
will  be  measured  by  the  half  of  BE\ 
the  angle  DAE  by  the  half  of  DE: 
hence,  their  difference,  BAD,  will  be 
measured  by  the  half  of  BE  minus  the 
half  of  ED,  or  by  the  half  of  BD. 
Hence,  every  inscribed  angle  is  measured 
by  half  the  arc  included  between  its  sides. 


I)  E 


74 


GEOMETEY. 


Cor.  1.  AU  tlie  angles  BAC,  BBC, 
BEC,  inscribed  in  the  same  segment 
are  equal ;  because  tliej  are  eacL. 
measured  by  half  of  the  same  arc 
BOC. 

Cor.  2.  Every  angle  BAB,  inscrib- 
ed in  a  semicircle  is  a  right  angle; 
because  it  is  measiu^ed  by  half  the 
semicircumference  BOB,  that  is,  by 
the  foui'th  part  of  the  whole  circum- 
ference (p.  17,  s.  2). 

-Cor.  o.  Every  angle  BAG,  inscrib- 
ed in  a  segment  greater  than  a  semi- 
circle, is  an  acute  angle;  for  it  is 
measured  by  half  the  arc  BOC,  less 
than  a  semicircumference  (p.  17,  s.  2). 

And    every  angle    BOC,  inscribed 
m  a  segment  less  than  a  semicircle,  is 
an  obtuse  angle  ;  for  it  is  measured  by   half  the   arc    BA  C, 
greater  than  a  semicuxumference. 

Cor.  -1.  The  opposite  angles  A  and 
C,  of  an  inscribed  quadrilateral  ABCD, 
ire  together  equal  to  two  right  angles  : 
for,  the  angle  BAB  is  measured  by 
half  the  arc  BCB,  the  angle  BCB  is 
measured  by  half  the  arc  BAB ;  hence, 
the  two  angles  BAB,  BCB,  taken  together,  are  measured 
by  half  the  circumference ;  hence,  their  sum  is  equal  to 
two  right  angles  (p.  17,  s.  2). 


PKOPOSITIOX  XIX.     THEOEEM. 


The  angle  formed  hy  two  chords,   ichich   intersect  each  other,  is 
measured  by  half  the  sum  of  the  arcs  included  between  its  sides. 

Let  AB,  CB,  be  two  chords  intersecting  each  other  at 
E:  then  will  the  angle  AEC,  or  BEB,  be  measured  by 
half  of  AC+DB. 


BOOK    III. 


75 


Draw  AF  parallel  to  DC:  the 
arc  DF  will  be  equal  io  AO  (p.  10), 
and  the  angle  FuiB  equal  to  the 
angle  DEB  (b.  l,  p.  20,  c.  3).  But  the 
angle  FAB  is  measured  by  half  the 
arc  FDB  (p.  18) ;  therefore,  DEB  is 
measured  by  half  of  FDB ;  that 
is,  by  half   of    DB+DF,    or  half  of 

dbX-ac, 

To  prove  the  same  for  the  angle  DEA,  or  its  equax 
BEC.  Draw  the  chord  AC.  Then,  the  angle  DCA  will 
be  measured  by  half  the  arc  DFA]  and  the  angle  BAC 
by  half  the  arc  CB  (p.  18).  But  the  outward  angle  AED^ 
of  the  triangle  EAC,  is  equal  to  the  sum  of  the  angles 
A  and  C  (b.  i.,  p.  25,  c.  6)  ;  hence,  this  angle  is  measured 
by  one-half  of  BC  plus  one-half  of  AFD ;  that  is,  by  half 
the  sum  of  the  intercepted  arcs.  By  drawing  a  chord 
BC,  similar  reasoning  would  apply  to  the  angle  AEC  or 
DEB, 


PEOPOSITION  XX.    THEOEEM. 


The   angle  formed  by    two    secants,    is    measured    hy  half  the 
difference  of  the  arcs  included  between  its  sides. 

Let  AB,  AC,  hQ  two  secants:  then  w'll  the  angle  BAC 
be  measured  by  half  the  difference  of  the  arcs  BEC  and 
DF. 

Draw  DE  parallel  io  AC:  the 
arc  EC  will  be  equal  to  DF  (p. 
10),  and  the  angle  BDE  equal  to 
the  angle  BAC  (b.  l,  p.  20,  c.  8). 
But  BDE  is  measured  by  half  the 
arc  BE  (p.  18) ;  hence,  BAC  is  also 
measured  by  half  the  arc  BE; 
that  is,  by  half  the  difference  of  BEC 
and  EC,  and  consequently,  by  half 
the  difference  of  BEC  and  DF. 


76 


GEOMETRY 


PEOPOSmOX    XXT.       TITEOEEM. 

Any  angle  formed  hy  a  tangent  and  a  chord  passing  through 

the  point  of  contact,  is  measured  by  half  the  arc  included 

between  its  sides. 

Let  BE  be  a  tangent,  and  AC  a  chord. 

From  A,  the  point  of  contact,  draw 
the  diameter  AD.  The  angle  BAB  is 
a  right  angle  (p.  9),  and  is  measured  by 
half  the  semicircumfereB.ce  AMD  (p.  17, 
s.  2) ;  the  angle  DA  C  is  measured  by  the 
half  oiDC]  hence,  BAD-VDA C,  or  BA G, 
is  measiu^ed  by  the  half  of  AJID  plus 
the  half  of  DC,  or  by  half  the  ^-hole 
arc  AJIDC 

It  may  be  shown,  by  taking  the  difference  of  the 
angles  DAB,  DAC,  that  the  angle  CAB  is  measured  by 
half  the  arc  AC,  included  between  its  sides. 


P  K  0  B  L  E  ^I  S 
RELATIXG   TO   THE    FIRST    AXD    THIRD    BOOKS 

PKOBLEM    I. 


To  bisect  a  given  straight  line. 

Let  AB  be  the  given  straight  line. 

From  the  points  A  and  B  as  cen- 
tres, with  a  radius  greater  than  the  half 
of  AB,  describe  two  arcs  cutting  each 
other  in  D]  the  point  D  will  be  equal- 
ly distant  from  A  and  B.  Find,  in  like  ^ 
manner,  above  or  beneath  the  hne  AB, 
a  second  point  B,  equally  distant  from 
the  points  A  and  B',  through  the  Irvvo 
points  D  and  E,  draw  the  line  DE, 
and  the  point  C,  where  this  line  meets  AB,  will  be  equally 
distant  from  A  and  B. 


><? 


B' 


BOOK   III.  77 

For,  tlie  two  points  D  and  E^  being  each  equally  dis- 
tant from  the  extremities  A  and  B^  must  both  lie  in  the 
perpendicular  raised  at  the  middle  point  of  AB  (b.  I.,  p.  16,  c). 
But  only  one  straight  line  can  be  drawn  through  two 
given  points  (a.  11) ;  hence,  the  line  BE  must  itself  be  that 
perpendicular,  which  divides  AB  into  two  equal  parts. 

PKOBLEM  II. 

At  a  given  pointy  in  a  given  straiglit  line,  to  erect  a  'perpendic- 
ular to  that  line. 

Let  BG  be  the  given  line,  and  A  the  given  point. 

Take  the  points  B  and  C  at  equal 
distances  from  A ;  then  from  the  points  /f 

B  and  C  as  centres,  with  a  radius  great- 
er  than   BA^  describe  two  arcs  inter- 
secting each  other  at  B ;  draw  AB  and       — %- 
it  will  be  the  perpendicular  required. 

For,  the  point  D,  being  equally  distant  from  B  and  (7, 
must  be  in  the  perpendicular  raised  at  the  middle  of  BG 
(b.  l,  p.  16) ;  and  since  two  points  determine  a  line,  AB  is 
that  perpendicular. 

Scholium.  The  same  construction  serves  for  making  a 
right  angle  BAB^  at  a  given  point  A^  on  a  given  straight 
line  BG. 

PEOBLEM   III. 

From  a  given  point,  ivithout  a  straight  line^  to  let  fall  a  per 
penclicidar  on  that  line. 

Let  A  be  the  point,  and  BB  the  given  straight  line. 

From  the  point  ^  as  a  centre,  and 
with  a  radius  sufficiently  great,  des- 
cribe an  arc  cutting  the  line  BB  in 
two   points   B   and  D;   then  mark  a     _\ 


O 


4-A 

/ 


point    E^    equally    distant    from    the       -^^^^ 
points   B  and  B^  and   draw   AE:   it  \l^, 

will  be  the  perpendicular  required. 

For,  the  two  points  A  and  E  are  each  equally  distant 
from  the  points  B  and  B\  hence,  the  line  AE  is  a  perpen- 
dicular passing  through  the  middle  of  BB  (b.  l.,  P.  16,  c). 


78  GEOMETET, 


PROBLEM  IV. 

At  a   point  in  a  given   line^  to  construct  an   angle  equal  to  a 

given  angle. 

Let  A  be  the  given  point,  AB  tlie  given  line,  and  ZZZ, 
the  given  angle. 

From  the  vertex  -ffj  as  a  -r  i)^ 

centre,  with    any  radius,  Kly 

K 


describe  the  arc   IL,  termina- 

ting  in  the  sides  of  the  angle.  I     A  B 

From  the  point  ^  as  a  centre,  with  a  distance  AB,  equal 
to  KI,  describe  the  indefinite  arc  BO;  then  take  a  radius 
equal  to  the  chord  ZTj  with  which,  from  the  point  ^  as  a 
centre,  describe  an  arc  cutting  the  indefinite  arc  BO,  in  JD; 
draw  AI) ;  and  the  angle  BAD  will  be  equal  to  the  given 
angle  K. 

For,  the  two  arcs  BD,  LI,  have  equal  radii,  and  equa! 
chords ;  hence,  thej  are  equal  (p.  4) ;  therefore,  the  angles 
BAD,  IKL,  measured  by  them,  are  also  equal  (p.  15). 

PEOBLEM   V. 

To  bisect  a  given  arc,  or  a  given  angle. 

First  Let  it  be  required  to  divide  the  arc  AEB  into 
two  equal  parts.  From  the  points  A  and  B,  as  centres, 
wi-th  equal  radii,  describe  two  arcs  cutting  each  other  in 
D;  through  the  point  D  and  the  centre  C,  draw  CD:  it 
wi]l  bisect  the  arc  AB  in  the  point  E. 

For,  the  two  points  C  and  D  are 
each  equally  distant  fi^om  the  extremi- 
ties A  and  B  of  the  chord  AB;  hence, 
the  line  CD  bisects  the  chord  at  right 
angles  (b.  i.,  p.  16,  c) ;  and  consequent- 
ly, it  bisects  the  arc  AEB  in  the  point 
E  (p.  6). 

Secondly.  Let  it  be  required  to  divide  the  angle  ACB 
into  two  equal  parts.  We  begin  by  describing,  from  the 
vertex  (7,  as  a  centre,  the  arc  AEB\   w^hich  is  tlien  bisect' 


BOOK    III.  79 

ed  as  above.     It  is  plain  tliat  the  line  CD  will  divide  tlie 
angle  ACB  into  two  equal  parts  (p.  17,  s.  1). 

Scholium.  By  the  same  construction,  each  of  the  halves 
AE^  EB^  may  be  divided  into  two  equal  parts ;  and  thus, 
by  successive  subdivisions,  a  given  angle,  or  a  given  arc, 
may  be  divided  into  four  equal  parts,  into  eight,  into  six- 
teen, and  so  on. 


PEOBLEM  VI. 

Through   a  given  pointy  to  draw   a   line  parallel  to  a  given 
straight  line. 

Let  A  be  the  given  point,  and  JBO  the  given  Kne. 

From  the  point  J.  as  a  centre, 
with  a  radius  ^^,  greater  than  the  ^' 
shortest  distance  from  A  to  BO, 
describe  the  indefinite  arc  EO ; 
from  the  point  ^  as  a  centre, 
with  the  same  radius,  describe  the  arc  AF;  lay  off  EI)= 
AF,  and  draw  AD:  this  is  the  parallel  required. 

For,  drawing  AE,  the  angles  AEF,  EAD,  aiO  equal 
(p.  15) ;  therefore,  the  lines  AD,  EF,  are  parallel  (b.  I., 
P.  19,  c.  1). 

PEOBLEM   VII. 

Two  angles  of  a  triangle  being  given,  to  find  the  third. 

Let  A  and  B  be  the  given  angles. 

Draw  the  indefinite  line  DEF\ 
at  any  point  as  E,  make  the  angle 
DEG  equal   to   the  angle  A,  and 
the  angle  GEH  equal  to  the  other 
angle  B\  the  remaining  angle  HEF 
will  be  the  third  angle  required;   because, these  three  angles 
are  together  equal  to  two  right  angles  (b.  i.,  p.  1,  c.  3),  and 
so   are    the   three   angles  of  a  triangle  (b.  I.,  P.  25) ;    conse- 
quently, HEF  is  equal  to  the  third  angle  of   the  triangle 


[F 

E  „ 

' 

^\ 

\  /-•'• 

^        \ 

\...'-^ 

\ 

A 

"i 

80 


GEOMETKY, 


PEOBLEM  Vin. 

Two  sides  of  a  triangle,  and  the  angle,  ivhicli  they  contain,  being 
given,  to  construct  the  triangle. 

Let  the  lines  B  and  G  be  equal  to  tlie  given  sides,  and 
A  tlie  given  angle. 

Having  drawn  the  indefinite 
line  DF^  make  at  the  point  D^ 
the  angle  FDE  eqnal  to  the  given 
angle  A  \  then  take  DG=B,  DH= 
Q  and  draw  GH  :  DGH  Avill  be 
the  triangle  required  (b.  I.,  p.  5). 


B 


PEOBLEM   IX. 

A  side  and  tioo   angles    of  a   triangle   being  given,  to  construct 

the  triangle. 

The  two  ansfles  will  either  be 
both  adjacent  to  the  given  side, 
or  one  will  be  adjacent,  and  the 
other  opposite :  in  the  latter  case 
find    the    third     angle     (peob.  7),  D  E 

and  the  two  adjacent  angles  will  be  known.  Then  draw 
the  straight  line  DB,  and  make  it  equal  to  .the  given  side: 
at  the  point  B,  make  an  angle  BBF,  equal  to  one  of  the 
adjacent  angles,  and  at  B,  an  angle  BBG  equal  to  the 
other ;  the  two  lines  BF,  BG,  will  intersect  each  other  in 
ZT;    and  BBH  will  be  the  triangle  required  (b.  I.,  p.  6). 


PEOBLEM    X. 

The  three  sides  of  a  triangle  being  given,  to  construct  the  triangle. 

Let  A,  B,  and  G,  denote  the  three  given  sides. 

Draw  BB,  and  make  it  equal 
to  the  side  A;  from  the  point  B 
as  a  centre,  with  a  radius  equal 
to  the  second  side  B,  describe  an 
arc;  from  ^  as  a  centre,  with  a 
radius  equal  to  the  third  side  G, 
describe  another  arc  intersecting  the  former  in  F  \  draw 
BF,  EF ;  and  DBF  will  be  the  triangle  required  (b.  I.,  P.  10). 


Bi- 


BOOK    III 


81 


Scholium,  If  one  of  the  sides  were  greater  than  the 
STim  of  the  other  two,  the  arcs  would  not  inteirsect  each 
other,  for  no  such  triangle  could  exist  (b.  i.,  p.  7):  but  the 
Lolution  will  always  be  possible,  when  the  sum  oi'each  two 
of  the  lines,  is  gi-eater  than  the  third. 


D 


PKOBLEM   XI. 

Two  sides   of  a   triangle^  and   the   angle   opposite   one  of  the7i\ 
being  given,  to  construct  the  triangle. 

Let  A  and  B  be  the  given  sides,  and  C  the  given  angle. 
There  are  two  cases. 

First.  When  the  angle  C  is 
a  right  angle,  or  when  it  is  ob- 
tuse. Draw  BF  and  make  the 
angle  FI)F=C]  take  I)F=A: 
from  the  point  ^  as  a  centre, 
with  a  radius  equal  to  the  given 
side  Bj  describe  an  arc  cutting 
DF  in  F-  draw  FF;  then  DFF 
will  be  the  triangle  required. 

In  this  case,  the  side  B  must 
be  greater  than  A  ;  for  the  angle  0  being  a  right  angle,  or 
an  obtuse  angle,  is  the  "greatest  angle  of  the  triangle  (b.  I., 
P.  25,  c.  3),  and  the  side  opposite  to  it  must,  therefore,  also 
be  the  greatest  (b.  i.,  p.  13). 

Secondly.  If  the  angle  0  is 
acute,  and  B  greater  than  A^  the 
same  construction  will  again  ap- 
ply, and  DFF  will  be  the  trian- 
gle required. 

lUit  if  the  angle  C  is  acute, 
and  the  side  B  less  than  A^  then 
the  arc  described  from  the  centre 
E,  with  the  radius  EF=B,  will 
cut  the  side  DF  in  two  points 
F  and  6^,  lying  on  the  same  side 
of  D :   hence,    there  will  be  two 

triangles   DFF,  DEG,   either  of    which  will  satisfy  all  the 
conditions  of  the  problem. 

6 


82  GEOMETRY. 

Scholium.  If  the  arc  described  with  ^  as  a  centre,  should 
be  tangent  to  the  line  BG,  the  triangle  would  be  right 
angled,  and  there  would  be  but  one  solution.  The  pro- 
blem will  be  impossible  in  all  cases,  when  the  side  B  is  less 
than  the  perpendicular  let  fall  from  B  on  the  line  BF, 

peoble:^!  XII. 

The  adjacent  sides  of  a  j^arcdklogram  and   their  included  angle 
being  given,  to  construct  the  jjarallelogram. 

Let  A  and  B  be  the  given  sides,  and  C  the  given  angle^ 

Draw  the  line  BH,  and 
lay  off  BE  equal  to  A  :  at 
the  point  1>,  make  the  angle 
EBF=C]  take  BF=B-  des- 
cribe two  arcs,  the  one  from 
F  as  a  centre,  with  a  radius 
FG=BF,  the  other  from  F 
as  a  centre,  with  a  radius  FG 
=BF ;  to  the  point  G,  where  these  arcs  intersect  each  other, 
draw  FG,  FG ;  BFGF  will  be  the  parallelogram  required. 

For,  the  opposite  sides  are  equal,  bj  construction; 
hence,  the  figure  is  a  parallelogram  (b.  I.,  P.  29);  and  it  is 
formed  with  the  given  sides  and  the  given  angle. 

Cor.  If  the  given  angle  is  a  right  angle,  the  figure  Anil 
be  a  rectangle ;  if,  in  addition  to  this,  the  sides  are  equal, 
it  will  be  a  square. 

PEOBLEM   XIII. 

To  find  the  centre  of  a  given  circle  or  arc. 

Take  three  points,  A,  B, 
C,  anywhere  in  the  circum- 
ference, or  in  the  arc ;  draAV 
AB,  BC,  or  suppose  them  to 
be  drawn;  bisect  these  two 
lines  by  the  perpendiculars 
BF,  FG  (PROB.  1) :  the  point 
0,  where  these  perpendicu- 
lars meet,  will  be  the  centre 
sought  (p.  6,  s). 


BOOK   III, 


83 


Scholium.  The  same  construction  serves  for  making  a 
circumference  pass  through  three  given  points  A^  i>,  (7; 
and  also  for  describing  a  circumference,  which  shall  cir- 
cumscribe a  given  triangle  ABC. 


PKOBLEM    XIV. 

Through  a  given  pointy  to  draio  a  tangent  to  a  given  circle. 

Let  A  be  the  given  point,  and  G  the  centre  of  the 
given  circle. 

A P 

If  the  given  point  A  lies  in  the 
circumference,  draw  the  radius  (7J.,  and 
erect  AB  perpendicular  to  it :  AI)  will 
be  the  tangent  required  (p.  9). 

If  the  point  A  lies  without  the  cir- 
cle, join  A  and  the  centre,  bj  the 
straight  line  CA :  bisect  CA  in  0 ; 
from  0  as  St.  centre,  with  the  radius 
00,  describe  a  circumference  intersect- 
ing the  given  circumference  in  B ; 
draw  AB :  this  will  be  the  tangent 
required. 

For,   drawing    CB,  the   angle    CBA 
being    inscribed    in   a   semicircle    is  a 
right  angle   (p.  18,  c.  2) ;   therefore,  AB  is  a  perpendicular  at 
the  extremity  of  the  radius  CB ;   hence,  it  is  a  tangent  (p.  9). 

Scholium  1.  When  the  point  A  lies  without  the  circle, 
there  will  be  two  equal  tangents,  AB,  AD,  passing  through 
the  point  A  :  for,  there  will  be  two  right-angled  triangles, 
CBA,  CBA,  having  the  hypothenuse  CA  common,  and  the 
eide  CB—CB;  hence,  there  will  be  two  equal  tangents, 
AB,  AD.     The  angles  CAB,  CAB,  are  also  equal  (b.  1.,  P.  17> 

Scholium  2.  As  there  can  be  but  one  line  bisecting  the 
angle  BAB,  it  follows,  that  the  line  which  bisects  the  angle 
formed  by  two  tangents,  must  pass  through  the  centre  of 
the  circle. 


8-1  GEOMETKY. 

PKOBLEM    XV. 

To  inscribe  a  circle  in  a  given  triangle. 

Let   ABC  be   tlie  given  triangle. 

Bisect  the  angles  A  and  B^ 
by  the  lines  AG  and  BO^ 
meeting  in  the  point  0  (prob. 
5);  from  the  point  0,  let  fiill 
the  perpendiculars  01),  OE^  OF 
(prob.  8),  on  the  three  sides  of 
the  triangle  :  these  perj^endic- 
iilars  will  all  be  equal. 

For,  by  construction,  we  have  the  angle  DAO^OAF, 
the  right  angle  ABO^AFO  \  hence,  the  third  angle  AOB 
is  equal  to  the  third  A  OF  (b.  i.,  p.  25,  c.  2).  Moreover, 
the  side  J.  0  is  common  to  the  two  triangles  A  OB^  A  OF ; 
and  the  angles  adjacent  to  the  equal  side  are  equal :  hence, 
the  triangles  themselves  are  equal  (b.  i.,  p.  6);  and  DO  is 
equal  to  OF.  In  the  same  manner  it  may  be  shown  that 
the  two  triangles  BOB.^  BOF^  are  equal  ;  therefore  OB  is 
equal  to  OE  \  hence,  the  three  perpendiculars  OB^  OE^  OF, 
are  all  equal. 

Now,  if  from  the  point  0  as  a  centre,  with  the  radius 
OB^  a  circle  be  described,  this  circle  will  be  inscribed  in 
the  triangle  ABC  (d.  11) ;  for,  the  side  AB^  being  perpen- 
dicular to  the  radius  at  its  extremity,  is  a  tangent  (p.  9); 
and  the  same  thing  is  true  of  the  sides  BC^  AC. 

Scholium.  The  three  lines  which  bisect  the  three  angles 
of  a  triangle  meet  in  the  same  point. 

PEOBLE^I     XVL 

On  a  given  straight  line  to  describe  a  segment  that  sJiall  contain 
a  given  angle;  that  is  to  sai/^  a  segment  such,  thai  any 
angle  inscribed  in  it  shall   be  equal  to  a  given  angle. 

Let  AB  be  the  given  straight  line,  and  C  the  given  angla 


BOOK    III. 


85 


Produce  AB  towards  D.  At  the  point  B,  make  the 
angle  JJBE=C;  draw  BO  perpendicular  to  BE\  and  at 
the  middle  point  G,  draw  GO  ])erpendicular  to  AB:  from 
the  point  0,  where  these  pcrjx^ndiculars  meet,  as  a  centre, 
with  the  distance  OB,  describe  a  circumference  :  the  re- 
quired segment  will  be  A  MB. 

For,  since  BF  is  perpendicular  to  the  radius  OB  at  its 
extremity,  it  is  a  tangent  (p.  9),  and  the  angle  ABF  is 
measured  by  half  the  arc  AKB  (p.  21).  Also,  the  angle 
AMB,  being  an  inscribed  angle,  is  measured  by  half  the 
arc  AKB  (p.  18) :  hence,  we  have  AMB=ABF=  EBD=C: 
hence,  any  angle  inscribed  in  the  segment  AJIB  is  equal 
to  the  given  angle   C. 

Scholium.  If  the*  given  angle  were  a  right  angle,  the 
required  segment  would  be  a  semicircle  described  on  AB  as 
a  diameter. 


PROBLEM   XVII. 

Two  angles  being  given,  to  find  their  common  measure,   if  they 
have  one,  and  hj  means  of  it,   their  ratio  in  numbers: 

Let  CAD  and  EBFhe  the  given  angles.  With  A  and  B 
as  centres,  and  with  equal  radii 
describe  the  arcs  CIJ,  EF,  to 
serve  as  measures  for  the  angles. 
Afterwards,  proceed  in  the  com- 
parison of  the  arcs  CB,  EF,  in 
the  same  manner  as  in  the 
comparison  of  two  straight  lines  (b.  II.,  D.  4) ;  since  an  arc 
may  be  cut  off  from  an  arc  of  the  same  radius,  as  a  straight 


86 


GEOMETKY. 


line  from  a  straight  line.  AVe 
shall  thus  arrive  at  the  com- 
mon measure  of  the  arcs  CD^ 
EF^  if  they  have  one,  and 
thereby  at  their  ratio  in  num- 
bers. This  ratio  will  be  the 
same  as  that  of  the  given  angles  (p.  17);  and  \i  DO  is  the 
common  measure  of  the  arcs,  the  angle  DAO  will  be  that 
of  the  angles. 

Scholium.  According  to  this  method,  the  absolute  value 
of  an  angle  may  be  found  by  comparing  the  arc  which 
measures  it,  with  a  quarter  circumference.  For  example,  if  a 
quarter  circumference  is  to  the  arc  CD  as  3  to  1,  then, 
the  angle  A  will  be  \  of  one  right  angle,  or  j^2  ^^  ^^'^ 
right  angles. 

It  may  also  happen,  that  the  arcs  compared  have  no 
common  measure  ;  in  which  case,  the  numerical  ratios  of 
the  angles  will  only  be  found  approximatively  with  more 
or  less  correctness,  according  as  the  oj^eration  is  continued 
ft  greater  or  less  number  of  times. 


BOOK    IV. 

PKOPOETIONS  OF  FIGUKES— MEASUREMENT  OF  AREAS 


DEFINITIONS. 


1.  Similar  Figures  are  those  which  are  mutually  equi- 
angular (b.  I.,  D.  22),  and  have  their  sides  about  the  equal 
angles,  taken  in  the  same  order,  proportional. 

2.  In  figures  which  are  mutually  equiangular,  the  angles 
which  are  equal,  each  to  each,  are  called  homologous  angles : 
and  the  sides  which  are  like  situated,  in  respect  to  the 
equal  angles,  are  called  homologous  sides. 

8.  Area,  denotes  the  superficial  contents  of  a  figure. 
The  area  of  a  figure  is  expressed  numerically  by  the  num- 
ber of  times  which  the  figure  contains  some  other  figure 
regarded  as  a  unit  of  measure. 

4,  Equivalent  Figures  are  those  which  have  equal 
areas.  The  term  egual^  when  applied  to  quantity  in  gen- 
eral, denotes  an  equality  of  measures;  but  when  applied 
to  geometrical  figures  it  denotes  an  equality  in  every  re- 
spect; and  such  figures  when  applied  the  one  to  the  other, 
coincide  in  all  their  parts  (a.  1-i).  The  term  equivalent^ 
denotes  an  equality  in  one  respect  only ;  viz. :  an  equality 
between  the  measures  of  figures.  The  sign  o,  denotes 
equivalency,  and  is  read,  is  equivalent  to. 

5.  Two  sides  of  one  figure  are  said  to  be  recip^vcally 
proj)ortional  to  two  sides  of  another,  when  one  of  the  sides 
of  the  first  is  to  one  of  the  sides  of  the  second,  as  the 
remaining  side  of  the  second  is  to  the  remaining  side  of 
the  first. 


88 


GEOMETRY. 


6.  SnriLAR  Arcs,  Sectors,  or.  Segments,  are 
which  ill  diftereut  circles,  correspond  to  equal  angles 
centre. 

Thus,  if  the  angles  A  and  0  are 
eqnal,  tlie  arc  BFC  will  be  similar  to 
DGE,  the  sector  BAC  to  the  sector 
DOF,  and  the  segment  BCF,  to  the 
segment  DBG. 

7.  The  Altitude  of  a  triangle  is  the 
perpendicular  let  fall  from  the  vertex  of 
an  angle  on  the  opposite  side,  or  on  that 
side  produced :  such  side  is  then  called 
a  base. 

8.  The  altitude  of  a  parallelogram 
is  the  perpendicular  distance  between 
two  opposite  sides.  These  sides  are 
called  ha-ses. 

9.  The  altitude  of  a  trapezoid  is  the 
perpendicular  distance  betvreen  its  two 
parallel  sides. 


those, 

at  the 


proposition  I.    theorem. 

Parallelograms  ichich   have  equal  bases  and  equal  altitudes,  are 

equivalent. 

Since  the  two  parallelograms  have  equal  bases,  those 
bases  may  be  placed  the  one  on  the  other.  Therefore,  let 
AB  be  the  common  base  of  the  two  parallelograms  A  BCD, 
ABEF,  which  have  the  same  altitude:  then  will  they  be 
equivalent. 

For,    in    the     parallelogTam     D      C  F       E    D  F    C   E 
ABCD,  we  have  \ 


B 


^i?=Z>qand^Z)=^C(B.i,P.28);        ^      ^ 

and  in  the  parallelogram  ABEF, 
we  have, 

AB  =  FF,  and  AF=BF: 
hence,  1)0= EF  (a.  1). 

Kow,  if  from  the  line  BF,  we  take  away  BC,  there  will 


BOOK    IV.  89 

remain  CE\    and  if  from  the  same  line  we  take  away  EF^ 
there  will  remain  DF] 
hence,  CE=DF  (a.  3) ; 

therefore,  the  triangles  ADF  and  BCE  are  mutually  equi- 
lateral, and  consequently,  equal   (B.  I.,  P.  10). 

But  if  from  the  quadrilateral  ABED,  we  take  away  the 
triangle  ADF,  there  will  remain  the  parallelogram  AIJEF] 
and  if  from  the  same  quadrilatend,  we  take  away  the  equal 
triangle  BCE,  there  will  remain  the  parallelograni  ABCD. 
Hence,  any  two  parallelograms,  which  have  equal  bases  and 
equal  altitudes,  are  equivalent. 

Scholium.  Since  the  rectangle  and  square  are  parallelo- 
grams (b.  I.,  D.  25),  it  follows  that  either  is  equivalent  to 
any  parallelogram  having  an  equal  base  and  an  equal 
altitude.  And  generally,  whatever  property  is  proved  as 
belonging  to  a  parallelogram,  belongs  equally  to  every 
variety  of  parallelogram. 

PKOrOSITION  11.      THEOREM. 

If  a  triangle  and  a  parallelogram  have  equal  bases  and  eguai 
altitudes,  the  triangle  luill  he  equivalent  to  half  the  par- 
allelogram. 

Place  the  base  of  the  triangle  on  that  of  the  parallelo- 
gram ABED:  then  will  they  have  a  common  base  AB. 

Now,  since   the  triangle  and  the    j)  F       E  G 

parallelogram    have   equal   altitudes,      V  \    /^        "y^ 

the    vertex   C.  of  the    triangle,   will         \  /  \  ^^  / 

be   in   the   upper    base    of   the  par-  ^\-l^^\./ 

allelogram,    or     in     that    base    pro-  A  B 

longed  (b.  I.,  P.  23).     Through  A,  draw  AE  parallel  to  BC^ 
forming  the  parallelogram  ABCE. 

Now,  the  parallelograms  ABED,  ABCE,  are  equivalent, 
having  the  same  base  and  the  same  altitude  (p.  1).  But  the 
triangle  ^Z>Cis  half  the  parallelogram  BE  (b.  i.,  p.  28,  c.  1): 
therefore,  it  is  equivalent  to  half  the  parallelouram  BB 
(A.  7). 

Cor.  All  triangles  which  have  equal  bases  and  equal 
altitudes  are  equivalent,  being  halves  of  equivalent  paral- 
lelograms. 


90 


GEOMETEY, 


TKOrOSITION  III.     THEOREM. 

5 16*0  Ttctamjles  having  equal  altitudes  are  to  each  other  as  their 

hases. 


D 


B 


Let  A  BCD,  AEFD,  be  two   rectangles  having  the  com 

mon  altitude  AB:  they  are  to  each  other  as  their  bases 
AB,  AE. 

Fit  St.  Suppose  that  the  bases 
are  commensurable,  and  are  to 
each  other,  for  example,  as  the 
numbers  7  and  -i.  If  AB  be  di- 
vided into  7  equal  parts,  AE 
will  contain  4  of  those  parts.  At  each  point  of  division 
erect  a  perpendicular  to  the  base  ;  seven  partial  rectangles 
will  thus  be  formed,  all  equal  to  each  other,  because  they 
have  equal  bases  and  the  same  altitude  (p.  1,  s).  The  rec- 
tangle A  BCD  will  contain  seven  partial  rectangles,  while 
AEFD  will  contain  four:  hence,  the  rectangle 

ABCD    :     AEFD    ::     7    :    4:,  or  txs  AB    :    AE. 

The  same  reasoning  may  be  applied  to  any  other  ratio 
equally  with  that  of  7  to  4 :  hence,  whatever  be  the  ratio, 
we  have,  when  its  terms  are  commensurable, 


ABCD 


AEFD 


AB 


AE. 
D 


FK   C 


Second.  Su]~)pose  that  the  bases  AB, 
AEj  are  incommensurable  :  we  shall  still 
have 

ABCD    :     AEFD    ::     AB    :     AE.  

ror,  it    the    rectangles  are  not  to  each 
other  in  the  ratio  of  AB  to  AE,  they  are  to  each  other  in 
a  ratio    greater   or    less :    that  is,  the    fourth    term  must  be 
greater   or   less   than   AE.      Suppose   it   to  be  greater,  and 
that  we  have 


ABCD 


AEFD 


AB    :     AO. 

Divide  the  line  AB  into  equal  parts,  each  less  than  EO. 
There  will  be  at  least  one  point  /  of  division  between  E 
and    0:    from   this   point   draw   IK  perpendicular    to   AIj 


BOOK    IV.  91 

forming  the  new  rectangle  AK:    then,  since  the  bases  AB^ 
AIj  are  commensurable,  we  have, 

ABCn    :     AIIfD    :  :     AB    :     AT. 
But  by  hypothesis  we  have 

ABCn    :     AEFD    :  :     AB    :     AO. 

In    these    two    proportions    the   antecedents    are   equal; 
hence,  the  consequents  aj-e  proportional  (b.  ii.,  p.  4),   that  is, 
AlKB    :     AEFD    :  :     AT    :     AO. 

But  AO  is  greater  than  AT;  which  requires  that  the 
rectangle  AEFD  be  greater  than  AiKD\  on  the  contrary, 
however,  it  is  less  (a.  8);  hence,  the  proportion  is  not  true; 
therefore  A  BCD  cannot  be  to  AFFD,  as  AB  is  to  a  line 
greater  than  AE. 

In  the  same  manner,  it  may  be  shown  that  the  fourth 
term  of  the  proportion  cannot  be  less  than  AE;  therefore; 
being  neither  greater  nor  less,  it  is  equal  to  AE.  Hence, 
any  two  rectangles  having  equal  altitudes,  are  to  each  other 
as  their  bases. 

PEOrOSITION    IV.     TnEOEEAl. 

Amj^two   rectangles   are   to   each   other  as  the  iwoducts  of  their 
hascs  CDid  altitudes. 

Let  A  BCD,  AEGF,  be  two  rectangles;  then  will  the 
rectangle, 

ABCD    :     AEGF    ::     ABxAD    :     AExAF. 

Having    placed    the    two  rect-      jt  p n 

angles,    so    that   the    angles  at  A 

are    oj^posite,    produce    the    sides 

GE,    CD,    till    they    meet    in    //. 

Then,  the   two  rectangles  ABCD, 

AEIID,  having   the  same  altitude      ^ 

AD,  are    to    each    other   as    their   bases  AB,  AE:    in   like 

manner  tlic  two  rectangles  AEIID,  AEGF,  having  the  same 

altitude  AE,  are    to    each    other    as    their    bases  AD,  AF- 

thus  we  have, 

ABCD    :     AEIID    :  :     AB    :     AE, 
AEIID   :     AEGF     ::     AD    :     AF, 


A 


B 


3i    1          1    1    1    i    ^    1 

2 

1 

i    1    1 

1 

•2 

3  1  4  j  5  i  6     7  1  8  •  9    10 

1)2  GEOMETRY. 

^[ultiplying  the  corresponding  terms  of  these  propor- 
tions together  (b.  il,  p.  13),  and  oinitting  the  factor  A  EHD, 
which  is  common  to  both  the  antecedent  and  coii sequent 
(b.  il,  p.  7),  we  have 

ABCD    :     AEGF    ::     ABxAD     :     AExAR 

Scholium  1.  If  we  take  a  Hne  of  a  given  length,  as  one 
inch,  one  foot,  one  vard,  &c.,  and  regard  it  as  the  linear 
unit  of  measure,  and  find  how  many  times  tliis  unit  is 
contained  in  the  base  of  any  rectangle,  and  also,  how  many 
times  it  is  contained  in  the  altitude  :  then,  the  product  of 
these  two  ratios  maj^  be  assumed  as  the  meaauro.  of  the 
rectangle. 

For  example,  if  the  base 
of  the  rectangle  A  contains 
ten  units  and  its  altitude  three, 
the  rectangle  will  be  repre- 
sented  by  the   number  10x3 

=30  ;    a  number  which   is   entirely  abstract,  so  long  as  we 
regard  the  numbers  10  and  3  as  ratios. 

But  if  we  assume  the  square  constructed  on  the  linear 
unit,  as  the  unit  of  surface,  then,  the  product  will  give 
the  number  of  superficial  units  in  the  surface ;  because,  for 
one  unit  in  height,  there  are  as  many  superficial  units  as 
there  are  linear  units  in  the  base;  for  two  units  in  height, 
twice  as  many ;  for  three  units  in  height,  three  times  as 
many,  &c. 

In  this  case,  the  measurement  which  before  was  merely 
relative,  becomes  absolute :  the  number  30,  for  example, 
by  which  the  rectangle  was  measui-ed,  now  rej^resents  30 
superficial  units,  or  30  of  those  equal  squares  described  on 
the  unit  of  linear  measure :  this  is  called  the  Area  of  the 
rectangle. 

Scholiinn  2.  In  geometry,  the  product  of  two  lines  fre- 
quentl}^  means  the  same  thing  as  their  redanrjle^  and  this 
expression  has  passed  into  arithmetic,  where  it  serves  to 
designate  th^  product  of  two  unequal  numbers.  The  term 
square  is  employed  to  designate  the  product  of  a  number 
multiphed  by  itsel£ 


BOOK    IV. 


93 


The  squares  of  tlie  numbers  1,  2,  8, 
&G.,  are  1,  4,  9,  &c.  So  likewise,  the 
geometrical  square  constructed  on  a 
double  line  is  evidently  four  times  as 
great  as  the  square  on  a  single  one ;  on 
a  triple   line   it  is   nine   times   as   great, 


PEOPOSITIUX  V.     THEOEEM. 

The  area  of  a  parallelogram  is  equal  to  the  2^'>'oduct  of  its  base 
and  altitude. 

Let  ABCD  be  any  parallelogram,  and  BE  its  altitude: 
then  will  its  area  be  equal  to  AB  X  BE.  Draw  BE  per- 
pendicular to  AB^  and  complete  the  rectangle  A  BEE. 

The    parallelogram    ABCD   is   equiv-       F      D       E       C 
alent    to    the    rectangle    ABEF  (p.  l,s.); 
but  this  rectangle  is  measured  by  ABx 
BE  (p.  4,  s.  V);    therefore,    ABxBE  is 
equal  to  the  area  of  the  parallelogram  ABCD. 

Cor.  Parallelograms  of  equal  bases  are  to  each  other  as 
their  altitudes;  and  parallelograms  of  equal  altitudes  are  to 
each  other  as  their  bases.  ^  For,  let  C  and  E  denote  the 
altitudes  of  two  parallelograms,  and  B  the  base  of  each : 

then,         BxC    :     BxD    ::     C    :     Z>  (b.  ii.,  p.  7), 

If  A  and  B  are  the  bases,  and  C  the  altitude  of  each,  we 
shall  have, 

AxC    :     BxC    ::     A     :     B: 

and  parallelograms,  generally,  are  to  each  other  as  the  pro- 
ducts of  their  bases  and  altitudes. 


PKOPOSITION    VI.     TUEOKEM. 

The  area  of  a  triangle  is  equal  to  half  the  product  of  its  hose 
and  altitude. 

Let  BAC  be  a  triangle,  and   AD   perpendicular   to   the 
base:   then  will  its  area  be  equal  to  one-half  of  BCxAD, 


94  GEOMETRY. 

For,  draw  CE  joarallel  to  BA^  and 
AE  parallel  to  BC,  completing  the  par- 
allelogram BE.  Then,  the  triangle  ABC 
is  half  the  parallelogram  ABCE,  which 
has  the  same  base  BC,  and  the  same 
altitude  AD  (p.  2) ;  but  the  area  of  the  parallelogram  is 
equal  to  BCxAD  (P.  6);  hence,  that  of  the  triangle  must 
be  IBCxAD,  or  BCx^AD. 

Cor.  Two  triangles  of  equal  altitudes  are  to  each  other 
as  their  bases,  and  two  triangles  of  equal  bases  are  to  each 
other  as  their  altitudes.  And  triangles  generally,  are  to 
each  other,  as  the  j^roducts  of  their  bases  and  altitudes. 

PKOPOSITION  VII.     TIIEOKEM. 

The  area  of  a   trapezoid  is  equal  to  tlie  product  of  its  altitude^ 
hij  half  the  sum  of  its  parallel  hoses. 

Let  ABCD  be  a  trapezoid,  EF  its  altitude,  AB  and  CB 
its  parallel  bases:  then  will  its  area  be  equal  to  EFX 
i{AB-{-Cn). 

Through  I,  the  middle  point  of  the  D  E      ^^      ^ 

side  BC,  draw  KL  parallel  to  the  op- 
posite side  AD]  and  produce  DC  till 
it  meets  KL. 


In  the  triangles  IBL,  ICK,  we  have      A       F  LB 

the  side  IB—IG,  by  construction;  the  angle  LIB=CIK 
(b.  I.,  P.  4) ;  and  since  CK  and  BL  are  parallel,  the  angle 
IBL—ICK  (b.  I.,  p.  20,  c.  2) ;  hence,  the  triangles  are  equal 
(b.  I.,  p.  6) ;  therefore,  the  trapezoid  ABCD  is  equivalent  to 
the  parallelogram  ALKD,  and  consequentl}^,  is  measured  by 
EFxAL  (p.  5). 

But  we  have  AL—DK\  and  since  the  triangles  IBL 
and  KCI  are  equal,  the  side  BL=CK'.  hence  AB+CD= 
AL-\-DK=2AL]  hence,  AL  is  the  half  sum  of  the  bases 
AB,  CD ;  hence,  the  area  of  the  trapezoid  ABCD,  is  equal 
to  the  altitude  EF  multiplied  by  the  half  sum  of  the  bases 
ABj  CD,  a  result  which  is  expressed  thus: 

ABi-CD 
ABCD=EFx Z 


BOOK    lY.  95 

Scholium.  If  tlirougli  Z  the  middle  point  of  BC,  the 
line  ///  be  drawn  parallel  to  the  base  AB^  it  Avill  bisect 
AD  at  //.  For,  since  the  figure  ALIII  is  a  parallelogram, 
as  also,  IIIKD^  their  opposite  sides  are  parallel,  and  we' 
\^YQ  AII  =  IL,  and  DTI=IK;  but  since  the  triangles  ZZ?./; 
fKC,  are  equal,  we  have  IL=IK;    therefore,  All =111). 

But  since  the  line  III=AL,  it  is  also  equal  to  ^ ; 

hence,  the  area  of  the  trapezoid  may  also  be  expressed 
Dy  EFxHI]  consequently,  tlie  area  of  a  trapezoid  is  equal 
to  its  altitude  multiplied  hy  tlie  line  which  connects  the  middle 
points  of  its  inclined  sides. 


rPwOPOSITION  VIII.     TIIECEEM. 

The  square  described  on  the  sum  of  two  lines  is  equivalent  to 
the  sum  of  the  squares  described  on  the  lines,  togdher  loith 
twice  the  rectangle  contained  by  the  lines. 

Let  AB^  BC^    be   any   two   lines,  and   AC   their   sum; 
then 

AG''  or  {AB-VBCf  o  AB^  +  W  4  2 .4J9  x  BO. 

On  AG  describe    the    square   AGDE ^    cake  AF=AB, 
draw  FG  parallel  to  AC^  and  BH  pardi^i  to  AE. 

The    square  AGDE    is    made    up   of       j^  IT        D 

four  parts ;  the  first  ABIF  is  the  square 
described  on  AB^  since  we  made  AF  = 
AB :  the  second  IGDII  is  the  square 
described  on  7(7,  or  BG  \  for,  since  we 
have  AG=AE   and   AB=AF,  the   dif- 


G 


ference,  AG—AB  must  be  equal  to  the      ^  B       C 

difference  AE—AF^  which  gives  BG  =  EF  \  but  IQ  is 
equal  to  BG^  and  DG  to  EF^  because  of  the  parallels; 
therefore,  IGDH  is  equal  to  a  square  described  on  BG 
Now,  if  these  two  squares  be  taken  away  from  the  large 
square,  there  will  remain  the  two  rectangles  BGGI, 
FIHE^  each  of  which  is  measured  by  ABxBG:  hence, 
the  square    on    the    sum   of   two    lines    is    equivalent    to 


m  GEOMETRY. 

the  sum  of  the    squares   on    tlie   lines,  together  with  t\vice 
the  rectangle  contained  by  the  lines. 

Cor.  If  the  line  AC  were  divided  into  two  equal  parts, 
the  two  rectangles  FH,  BG,  would  become  squares,  and  the 
square  described  on  the  whole  line  would  be  equivalent  to 
four  times  the  square  described  on  half  the  line. 

Scholium,  This  property  is  the  same  as  the  property 
demonstrated  in  algebra,  in  obtaining  the  square  of  a 
binomial ;    which   is   expressed  thus : 


PEOPOSITIOX  IX.     TIIEOEEM. 

Fne  square  described  on  the  difference  of  two  lines,  is  equivalent 
to  the  sum  of  the  squares  described  on  the  lines,  diminished 
hy  ticice  the  rectangh  contained  by  the  lines. 

Let   AB^  BC,    be   two   lines,  and   AC  their   difference; 
then,  .!(;',  or  {AB-BCf --^Alf-\-BC'-2ABxBC. 

On  AB  describe  the  square  ABIF;      L__F G I 

tale  AE=AC:  tlirougli  (7  draw  CG 
parallel  to  BI,  and  through  E  draw 
EI£  parallel  to  AB,  and  prolong  it 
to  K,  making  FK=CB,  and  then 
complete  the  square  KFFL.  L 1 — . 

Since  KD=AB,  and  BC=KL,  the 
two   rectangles  CI,  KG,   are   each   measured   by  ABxBC: 

the  whole  figure  ABILKEA,  is  equivalent  to  AB%BC'' ; 
take  from  each  the  two  rectangles  CI,  KG,  and  there  will 
remain  the  square  ACDE,  equivalent  to  AB'-^-BC  dimin- 
ished by  twice  the  rectangle  of  ABxBC. 

Scholiu7yi.  This  property  is  expressed   by  the  algebraical 
formula, 

(^a-bf^a''-2ab+b\ 


BOOK   IT.  97 


PKOPOSITION  X.     THEOEEM. 


The  rectangle   contained   hy   the  sum  and   the  difference  of  two 
lines,  is  equivalent  to  the  difference  of  their  squares. 


Let  AB^  BO^  be  two  lines ;   tlieii 


F                G     I 

E 

H 

D 

{AB+BG)X{AB-BG)^^AB  -BG\ 

Upon  AB  and  AC,  describe  tlie 
squares  ABIF^  ACBE\  prolong  AB 
till  BK  is  equal  to  ^C;  and  com- 
plete the  rectangle  AKLE,  and  pro- 
long CD  to  a. 

The  base  AK  of  the  rectangle 
AL  is  the  sum  of  the  two  lines  AB 

BC\  and  its  altitude  AE  is  their  difference ;   therefore,  the 
rectangle  AKLE  is  equivalent  to 

{AB-^BC)X{AB-BC). 

Again,  DHIG  is  equal  to  a  square  described  on  CB ; 
and  since  BH  is  equal  to  ED^  and  BK  to  EF^  the  rect- 
angle BL  is  equal  to  the  rectangle  EG :  hence,  the  rect- 
angle AKLE  is  equivalent  to  ABHE  plus  EDGF^  which 
is  precisely  the  difference  between  the  two  squares  AL  and 
DL  described  on  the  lines  AB,  CB :   hence,  we  have  (a.  1.), 

{AB-\-BC)x{AB-BG)=o^AB'-BG''. 

Scholium.  This  property  is  expressed  by  the  algebraical 
formula, 

{a^-h)X{a-h)=a''-lJ'. 

PKOPOSITION  XI.     THEOEEM. 

The  sqtiare  described  on  the  hypotlienuse  of  a  right-angled  tri 
angle  is  equivalent  to  the  sum  of  the  squares  described  on  the 
other  two  sides. 

Let  BGA  be  a  right-angled  triangle,  right-angled  at  A: 
then  will  the  square  described  on  the  hypothenuse  BG  be 
equivalent  to  the  sum  of  the  squares  described  on  the  other 
two  sides,  BA,  AG. 


^ 


GEOMETRY. 


Having  described  a  square 
on  eacli  of  the  three  sides, 
let  fall  from  A,  on  the  hy- 
j)0thenuse,  the  perpendicular 
AD,  and  prolong  it  to  F] 
nd  draw  the  diagonals  AF^ 
CIL 

The  angle  ABF  is  made 
up  of  the  angle  ABC^  to- 
gether with  the  right  angle 
CBF]  the  angle  CBHis  made 
up  of  the  same   angle  ABC^ 

together  with  the  right-angle  ABR-  hence,  the  angle  ABF  is 
equal  to  RBC  (a.  2).  But  we  have  AB=-BH,  being  sides  of 
the  same  square;  and  BF=BC.  for  the  same  reason:  there- 
fore, the  triangles  ABF^  HBO^  have  two  sides  and  the  in- 
cluded angle  equal,  each  to  each ;  therefore,  they  are  them- 
selves equal  (b.  i.,  p.  5). 

But  the  triangle  ABF  is  equivalent  to  half  the  rectangle 
BE^  because  they  have  the  same  base  BF^  and  the  same 
altitude  BD  (p.  2).  The  triangle  HBO^  in  like  manner  is 
equivalent  to  half  the  square  AH:  for,  the  angles  BAC^ 
BAL,  being  both  right  angles,  AC  and  AL  form  one  and 
the  same  straight  line  parallel  to  HB  (b.  i.,  p.  3) ;  hence, 
the  triangle  and  square  have  equal  altitudes  (b.  I.,  P.  23) ; 
they  also  have  the  common  base  BH;  consequently,  the 
triangle  is  half  the  square  (p.  2). 

The  triangle  ABF  has  already  been  proved  equal  to  the 
triangle  IIBC:  hence,  the  rectanole  BDEF,  which  is  double 
the  triangle  ABF,  must  be  equivalent  to  the  square  AH, 
which  is  double  the  equal  triangle  ffBC.  In  the  same 
manner  it  may  be  proved,  that  the  rectangle  EG  CD  is  equiva- 
lent to  the  square  AL  But  the  two  rectangles  FEDB^  EG  CD, 
taken  together,  make  up  the  square  FG  CB :  therefore,  the 
square  FGCB,  described  on  the  hypothenuse,  is  equivalent 
to  the  sum  of  the  squares  BALH^  CIKA,  described  on  the 
two  other  sides ;   that  is, 

BC'<:=ABr+AC', 


Co'^.  1.   Hence,  the  square  of  one  of  the  sides  of  a  right- 


BOOK   lY.  99 

angled  triangle  is  equivalent  to  the  square  of  the  hypothe- 
nuse  diminished  by  the  square  of  the  other  side ;   thus, 

Cor.  2.  If  from  the  vertex  of  the  right  angle,  a  perpen 
iicular  be  let  fall  on  the  hjpothenuse,  the  parts  of  the 
hypothenuse  are.  called  segments:  we  shall  then  have, 

The  square  of  the  hypothenuse  is  to  the  square  of  either  side 
about  the  right  angle,  as  the  hypothenuse  to  the  segment  adjacent 
to  that  side. 

For,  by  reason  of  the  common  altitude  BF,  the  square 
BG  is  to  the  rectangle  BE,  as  BC  to  BD  (p,  3) :  but  the 
square  BL  is  equivalent  to  the  rectangle  BE:   hence 

BG"    :    BA"    ::     bo    :     BJD. 

We  may  show,  in  like  manner,  that 

BO"    :    AC''    ::    BG    :    DC. 

Cor.  3.  The  squares  of  the  two  sides  containing  the  right 
angle,  are  to  each  other  as  the  adjacent  segments  of  the  hypothe- 
nuse. 

For,  the  rectangles  BDEF,  DCGE,  having  the  same 
altitude,  are  to  each  other  as  their  bases  BD,  CD  (p.  3). 
But  these  rectangles  are  equivalent  to  the  squares  AH,  AT] 
therefore,  we  have 

11?    :     AG^    ::    BD    :     DC. 

Cor.  4.  The  square  described  on  the  diagoncd  of  a  square 
is  equivalent  to  double  the  square  described  on  a  side. 

Let  ABCD  be  a  square  described  on 
AB,  and  EFOH  a  square  described  on 
the  diagonal  AG.  The  triangle  ABC 
being  right-angled  and  isosceles,  we  shall 
have 

J^KD=  AB''-\-BG'=o=  2AB\ 

It  is  plain,  that  of  the  eight  equal  right-angled  triangles 
which  compose  the  square  EG,  four  will  lie  without  the 
square  ABCD,  and  four  within  it :  hence,  the  square  on  the 
diagonal  is  equivalent  to  double  ilie  square  on  the  side. 


100 


GEOMETRY, 


(hr.  (;,   Bj  tlie  last  corollarj,  we  Lave 
AO"    :    AB"    :  :     2     :     1  ; 
hence,  by  extracting  tlie  square  root  (b.  ii.,  p.  12,  c), 
AC    :    AB     ^/'2    '.     1:      - 

tliat  is,  the  diagonal  of  a  square  is  to  the  side  as  tlie  square 
root  of  two  to  one :  consequently,  the  diagonal  and  side  of  a 
square  are  ijicoinmensurable. 


PKOPOSITION  XII.    THEOEEM. 

In  any  triangle,  the  square  of  a  side  ojyposite  an  acute  angle  is 
equivalent  to  tlie  sum  of  the  squares  of  the  base  and  Hie 
other  side,  diminished  hy  twice  the  rectangle  contained  hy  the 
base  and  the  distance  from  the  vertex  of  the  acute  angle  to 
the  foot  of  the  perpendicular  let  fall  from  the  vertex  of  the 
opposite  angle  on  the  base,  or  on  the  base  produced. 

Let  ABC  be  a  triangle,   C  one  of  the  acute  angles,  and 
AD  perpendicular  to  the  base  EC',   then  will 

AK=<y=BG'-\-AC^-2BCx  GD. 

First.  When  the  perpendicular  falls 
within  the  triangle  ABC,  we  have  RD= 
BC—CD,  and  consequently, 

Bff=o=W^-\-CIf-2BCxCD  (p.  9). 

Adding  AB"  to  each,  and  obser\'ing  that 
the    right-angled    triangles  ABB,   ABC, 

give         AB-^BB^oAK,  and  AB-^-CffoA^, 

we  have  AB'oBC'-\-AB-2BCx  CD. 

Secondly.  When  the  perpendicular  AB 
falls  without  the  triangle  ABC,  we  have 
BB=CB—BC ',    and    consequently, 

BB'=o  CB^BC'-2CDxBC  (p.  9). 

Adding  AB  to  both,  we  find,  as  before, 

AB'o=W-\-AB-2BCx  CD. 


D     B 


BOOK    lY.  101 


PROPOSITION  XIII.     THEOEEM. 

Tn  any  ohtuse-angled  triangle,  the  square  of  the  side  opposite  the 
obtuse  angle  is  equivalent  to  the  sum  of  the  squares  of  the 
base  and  the  other  side,  augmented  by  twice  the  rectangle 
contained  by  the  base  and  the  distance  from  the  vertex  of  the 
obtuse  angle  to  the  foot  of  the  perpendicular  let  fall  from  the 
vertex  of  the  opposite  angle  on  the  base  produced. 

Let  ACB  be  a  triangle,  G  the  obtuse  angle,  and  AB 
perpendicular  to  BO  produced ;   then 

Z^=o=la>^V2^(7x  CD, 
For,  we  have,  BD=BG-\-CD; 
consequently  (p.  8), 

BlfoBG\CIX-\-2BGx  GD. 

Adding  AD^  to  both  members,  and  reducing  as  in  the  last 
theorem,  and  we  have 

AB'^=<^BG^^+AG^+2BGx  GD. 

Scholium.  The  right-angled  triangle  is  the  only  one  in 
which  the  sum  of  the  squares  described  on  two  sides  is 
equivalent  to  the  square  described  on  the  third;  for,  if  the 
angle  contained  by  the  two  sides  is  acute,  the  sum  of  their 
squares  is  greater  than  the  square  of  the  opposite  side;  ii 
obtuse,  it  is  less. 


PEOPOSITION  XIV.     THEOEEM. 

In  any  triangle,  the  sum  of  the  squares  described  on  two  sides 
is  equivalent  to  twice  the  square  of  half  the  third  side,  plus 
twice  the  squ/xre  of  the  line  drawn  from  the  middle  point  of 
that  side  to  the  vertex  of  the  opposite  angle. 

Let  ABG  be  any  triangle,  and  AE  a  line  drawn  to  the 
middle  of  the  base  BG  \    then 


102  GEOMETKY. 

For,  on  BC,  "let  fall  the  perpendic- 
ular AD.     Then, 

AG^^oAK+EC'-2ECxED.  (p.  12). 

And,  B  ElD       C 

Ib''=oAE'+EB'^+2EBxED  (p.  13). 

Hence,    bj    adding    and    observing   that   EB  and  EC  are 
equal,  Ave  have 

J:B"+ JL^"=0=  2Z5 '+2  Jif . 

(7or.  1.   In  any  quadrilateral,  the  sum  of  the  squares  of  the 
four  sides  is  equivalent   to  the  sum  of  the  squares  of  the  two 

diagonals,  plus  four    times    the   square  of  the  line  joining  tJie 
middle  points  of  the  diagonals. 

Let  ABCD  be  a  quadrilateral,  J.CJ 

BD,  the  diagonals,  and  EF  a  line  join-  R~-___^P 

ing  their  middle  points  E  and  F.  \  \  /  \\ 

From  the  theorem,  we  have  /e1/---!-vA 


aD>  C5^=o=  2^i^'+2  of" 


AV-^AB'-Q^IBF'^IAF'  • 
and  from  the  same  theorem,  by  multipljing  by  2, 

2GF^-V1AF~=^\AE'-V\EF^^  : 
hence,  by  addition, 

whence  (p.  8,  c), 

CIf-\-  CB'^AL'+AB'=c^BD'+AC'+4:  ET. 

Cor.  2.  In  the  case  of  the  parallelogTam  the  points  E  and 
F  will  coincide,  and  the  sum  of  the  squares  described  on  the 
sides  will  be  equivalent  to  the  sum  of  the  squares  describ- 
ed on  the  diagonals. 


BOOK    IV. 


103 


PROPOSITION  XV.     THEOREM. 


If^  in  any  triangle^  a  line    he    draivn  parallel    to    the    hasc^  it 
will  divide  the  two  other  sides  loroijortionalhj. 


straiglit  line  drawD 


EC. 


Let  ABC  be  a  triangle,  and  DE  a 
parallel  to  the  base  BC  ]   then 

AD    :     DB    ::     AE    : 

Draw  the  lines  BE  and  CB.  Then, 
the  triangles  ABE,  BBE,  having  a 
common  vertex,  E,  have  the  same  alti- 
tude, and  are  to  each  other  as  their 
bases  (p.  6,  c.) ;  hence  we  have 
ABE    :     BBE    :  :     AB    :     BB. 

The  triangles  ABE,    BEC,    with    a 
common  vertex  B,  also  have  the  same  altitude,  and  are  to 
each  other  as  their  bases ;    hence, 

ABE    :     BEC    :  :     AE    :     EC. 

But  the  triangles  BBE,  BEC,  are  equivalent,  having  the 
same  base  BE,  and  their  vertices  B  and  (7  in  a  line  paral- 
lel to  the  base :    and  therefore,  we  have  (b.  ii.,  p.  4,  c.) 
AB    :     BB    ::     AE    :     EC 

Cor.  1.   Hence,  by  composition,  we  have  (b.  ii.,  p.  6), 
AB+BB  :AB::  AE+EC  :  AE,  or  AB  :  AB::  AC:  AE; 
and  also,  AB  :   BB  :  :   AC  :    CE. 

Cor.  2.  If  any  number  of  parallels  AC,  EF,  GH,  BB, 
be  drawn  between  two  straight  lines  AB,  CB,  those 
straight  lines  will  be  cut  proportionally,  and  we  shall  have 

AE    :     CF    ::     EG    :    FH    :     QB    :    HB. 
For,  let  0  be  the  point   where  AB 
and  CB  meet.     In   the   triangle  GEE, 
the   line  AC  being   drawn   parallel   to 
the  base  EF^  we  shall  have 

GE    :    AE    ::     GF    :     CF. 
In    the    triangle    GGH,    we   shall  like- 
wise have 

GE    :     EG    ::     GF    :     FlI. 


iO-i 


GEOMETRY. 


And,  by  reason  of  the  common  ante- 
cedents OE^   OF  (b.  II.,  p.  4),   we  have 

AE    :     CF    ::     EG    :     FH. 

It  may  be  proved  in  the  same  manner, 
hat 

EG    :     FH    ::     GB    :     ED, 

and  so  on ;  hence,  the  lines  AB^  CD^ 
are  cut  proportionally  by  the  parallels 
AG,  EF,  GH,  <fcc. 


PEOPOSITION  XVI.     THEOEEM. 


If  two  sides    of  a  triangle  ara-  cut  proportionally    hy  a  straight 
line,  this  straight  line  icill  he  parallel  to  the  third  side. 

In  the  triangle  BA  G,  let  the  line  BE  be  drawn,  cutting 
the  sides  BA  and  GA  proportionally  in  the  points  B  and  E\ 
that  is,  so  that 

BB    :    BA    '.:     GF    '.     EA'. 
then  will  BE  be  parallel  to  BG. 

Having   drawn   the    lines   BE  and 
BG,  we  have   (p.  6,  c), 


BBE    :     BAE    :: 

BB    : 

BA, 

BEG    :     BAE    :  : 

GE    : 

EA: 

but,  by  hypothesis, 

BB    '.     BA    '.'. 

GE    : 

EA: 

hence  (b.  II.,  P.  4,  c), 

BBE    :     BAE    :  : 

BEG 

:     BA 

B 


and  since  BBE  and  BEG  have  the  same  ratio  to  BAE^ 
they  have  the  same  area,  and  hence  are  equivalent  (d.  4). 
They  also  have  a  common  base  BE;  hence,  they  have  the 
same  altitude  (p.  6,  c.) ;  and  consequently,  their  vertices  B 
and  G  lie  in  a  parallel  to  the  base  BE  (b.  I.,  P.  23) :  hence, 
DE  is  parallel  to  BG. 


BOOK    lY. 


105 


PKOPOSITION  XVII.     TIIEOEEM. 


The  line  which  bisects  the  vertical  angle  of  a  triangle^  divides 
the  base  into  two  segments^  which  are  proportional  to  the 
adjacent  sides. 


AG. 


In  the  triangle  ACB^  let  AD  be  drawn,  bisecting  the 
angle  CAB]   then 

BD    :     CD    ::     AB 

Through  the  point  C  draw  -g 
GE  parallel  to  AD^  and  prolong  \- 
it  till  it  meets  BA  produced  in  E.       \ 

In  the  triangle  BCE^  the  line 
AD  is  parallel  to  the  base  CE  \ 
hence,   we  have  the    proportion 
(P.  15), 
BD    '.    DC    ','.    BA    '.    AE. 

But  the  triangle  ACE  is  isosceles:  for,  since  AD^  GB^ 
are  parallel,  we  have  the  angle  ACE—DAG^  and  the  angle 
AEC=BAD  (b.  l,  p.  20,  c.  2,  3) ;  but,  by  hypothesis,  DAG 
=DAB]  hence,  the  angle  ACE— AEG,  and  consequently, 
AE=AG  (b.  I.,  p.  12).  In  place  of  AE  in  the  above  pro- 
portion,  substitute  AG^  and  we  shall  have, 

BD    '.    DC    :-.    AB    :    AC 

Cor.  If  the  line  AD  bisects  the  exterior  angle  GAE  of 
the  triangle  BAC^  we  shall  have, 

BD    ',     CD    '.    AB 

For,  through  G  draw  GF  par- 
allel to  AD. 

Then,  the  angle  GAD=AGF, 
and,  the  angle  EAD=AFC] 

hence,  (a.  1),  the  angle  ACF=AFG] 
consequently,  AF  is  equal  to  A  C 

But,  since  EC  is  parallel  to 
the  base  AD^ 

BD    :     DC    :     AB    :     AF] 
hence,  BD    :    DC    :    AB    :    AG. 


AG 


106  GEOMETEY 


PEOPOSITIOX  XVIU.     THEOKEM. 


Equiangular  bnangles   have   their  homologous  sides  proporttoritH 
and  are  similar. 

Let  BCA  and  CED  be  t^vo  equi- 
angular triangles,  having  the  angle 
BAC=CDE,^ABC=DCE,  and  ACB 
=DEG ;  then,  the  homologous  sides 
wall  be  proportional,  "s4z. : 

BC    :     CE    ::     BA    :     CD    w 

Place  the  homologous  sides  BC,  CE  in  the  same  straight 
line ;    and  prolong  the  sides  BA^  ED,  till  thej  meet  in  F. 

Since  BCE  is  a  straight  line,  and  the  angle  BCA  equal 
to  CED,  it  follows  that  AC  \s>  parallel  to  DE  (b.  I.,  P.  19, 
c.  2).  In  hke  manner,  since  the  angle  ABC  is  equal  to 
DCE^  the  line  AB  is  parallel  to  DC.  Hence,  the  figure 
ACDF  is  a  parallelogram,  and  has  its  opposite  sides  equal 
(B.  L,  P.  28). 

In  the  triangle  BEE,  the  line  AC  is>  parallel  to  the  base 
FE]   hence,  we  have  (p.  15.) 

BC    :     CE    ::     BA     '.     AF; 

or  putting  CD  in  the  place  of  its  equal  AF, 

BC    :     CE    ::     BA    :     CD. 

In  the  same  triangle  BEE,  CD  is  parallel  to  BE;  ard 
Lenc^, 

BC    :     CE    ::     FD    :     DE] 

or  putting  AC  in  the  place  of  its  equal  FD, 

BC    :     CE    ::     AC    :    DE. 

And  finally,  since  both  these  proportions  have  an  ante- 
cedent and  consequent  common,  we  have  (b.  il,  p.  4,  c), 

BA    :     CD    ::    AC    :     DE. 

Thus,  the  equiangular  triangles  CAB^  CED^  have  their 
homologous  sides  proportional.  But  two  figures  are  similar 
when  they  have  their  angles  equal,  each  to  each,  and  their 


BOOK  lY.  107 

homologous  sides  proportional  (d.  1,   2);    consequently,    the 
two  equiangular  triangles  BA  (7,   CED^  are  similar  figures. 

Cor.  Two  triangles  which  have  two  angles  of  the  one 
equal  to  two  angles  of  the  other,  are  similar ;  for,  the  third 
angles  are  then  equal,  and  the  two  triangles  are  equian 
gular  (b.  I.,  p.  25,  c.  2.) 

Scholium.  Observe,  that  in  similar  triangles,  the  homolo- 
gous sides  in  each  are  opposite  to  the  equal  angles;  thus, 
the  angle  BOA  being  equal  to  CJED,  the  side  AB  is  homo- 
logous to  DC;  in  like  manner  AG  and  BE  are  homologous, 
because  opposite  to  the  equal  angles  ABO,  DOE. 

PROPOSITION  XIX.     THEOKEM. 

Conversely :    Triangles^  ichich  have   their  sides  proportional,  are 
equiangular  and  similar. 

If,   in  the  two  triangles  BAC,  EBF,  we  have, 

BC    :     EF    ::     BA    :    ED    ::    AC    :     DF -, 

then  will  the  triangles  BAC,  EBF,  have  their  angles  equal, 
uamelj, 

A^B,  B=E,   C=F. 

At  the  point  E,  make  the  angle 
FEG=B,  and  at  F,  the  angle  EFG 
=  (7;  tlie  third  angle  G  will  then 
be  equal  to  the  third  angle  A  (b. 
[.,  p.  25,  c.  2).  Therefore,  by  the 
last  theorem,  we  shall  have 

BG  :  EF  ::  AB  :  EG : 
but,  by  hypothesis,  we  have 

BC  :  EF  ::  AB  :  BE; 
hence,  EG=BE.     By  the  same  theorem,  we  shall  also  have 

BC  :  EF  i:  AC  :  FG; 
and  by  hypothesis,  we  have 

BO  :  EF  ::  AC  :  BE-, 
hence,  FG=BF,    Hence,  the  triangles  EGF,  FED^  having 


108 


GEOMETKY 


their  three  sides  equal,  each  to 
each,  are  themselves  equal  (b.  I., 
p.  10).  But,  by  construction,  the 
triangles  EGF  and  ABC  are  equi- 
anorular:  hence,  DEF  and  ABC 
are    also   equiangular    and   similar 

(A.  1). 

Scholium  1.  By  the  last  two  propositions,  it  appears  that 
triangles  which  are  equiangular  are  similar:  and  conversely: 
if  triangles  have  their  sides  jDroportional,  they  are  equiangu- 
lar, and  consequently,  simiLar. 

The  case  is  different  ^\\X\  regard  to  figures  of  more  than 
three  sides :  even  in  quadrilaterals,  the  proportion  between 
the  sides  may  be  altered  without  changing  the  angles,  or 
the  angles  may  be  changed  without  altering  the  proportion 
between  the  sides.  Thus,  in  quadrilaterals,  equality  between 
the  corresponding  angles  does  not  insure  proportionality 
among  the  sides :  and  reciprocally :  proportionality  among 
the  sides  does  not  insure  equality  among  the  corresponding 
angles.  It  is  evident,  for  example,  that 
if  in  the  quadrilateral  ABCD^  we  draw 
EF  parallel  to  BC,  the  angles  of  the 
quadrilateral  AEFD,  are  made  equal  to 
those  of  ABCD ;  though  the  proportion 
between  their  sides  is  different;  and  in 
like  manner,  without  changing  the  four 
sides  AB^  BC^  CD,  AD,  we  can  change  the  angles  by 
making  the  point  B  approach  to  D,  or  recede  from  it. 

Scholium  2.  The  two  preceding  propositions,  are  in  strict- 
ness but  one,  and  these,  together  with  that  relating  to  the 
square  of  the  hypothenuse,  are  the  most  important  and 
fertile  in  results  of  any  in  geometry.  They  are  almost 
sufficient  of  themselves  for  every  application  to  subsequent 
reasoning,  and  for  solving  every  problem.  The  reason  is, 
that  all  figures  may  be  divided  into  triangles,  and  any  tri- 
angle into  two  right-angled  triangles.  Thus,  the  properties 
of  triangles  include,  by  implication,  those  of  all  figuj-es. 


BOOK   lY 


109 


PKOPOSITION  XX.     THEOKEM. 


Two  triangles,  which  have  an  angle  of  the  one  equal  to  an  angle 
of  the  other,  and  the  sides  containing  those  angles  proporticma\ 
.  are  similar. 


DR 


Let  ABC,  BEF^  be   two  triangles,  having  the  angle  A 
equal  to  B\   then,  if 

AB    :    BE    '.'.    AC 
the  two  triangles  will  be  similar. 

Make  AG=BE,  and  draw  GH 
parallel  to  BG.  The  angle  AGE  will 
be  equal  to  the  angle  ABC  (b.  i.,  p. 
20,  c.  8);  and  the  triangles  AGE, 
ABC,  will  be  equiangular:  hence,  we 
shall  have, 

AB    :     AG    :       AG 
But,  bj  hypothesis,  we  have, 

AB  :  BE  ::  AG  :  BF \ 
and  by  construction,  AG—BE'.  hence  AE—BF.  There- 
fore, the  two  triangles  AGE,  BEF,  have  two  sides  and  the 
included  angle  of  the  one  equal  to  two  sides  and  the  in- 
cluded angle  of  the  other :  hence,  they  are  equal  (b.  i.,  p.  5) ; 
but  the  triangle  AGE  is  similar  to  ABC :  therefore,  BEF 
is  also  similar  to  ABO. 


PEOPOSITION  XXI.     THEOEEM. 

Two  triangles,  which  have  their  sides,  iivo  and  two,  either  par- 
allel  or  perpendicular  to  each  other,  are  similar. 

Let  BAG,  EBF,  be  two  triangles,  having  their  sides  re- 
spectively parallel  to  each  other ;  then  will  they  be  similar. 

First.  If  the  side  BA  is  parallel  to 
EB,  and  BC  to  EF,  the  angle  ABC 
is  equal  to  BEF  (b.  l,  p.  2-i) :  if  CA 
is  parallel  to  FB,  the  angle  BCA 
is  equal  to  FEB,  and  also,  BAG  to 
EBF',  hence,  the  triangles  GBA, 
FEB,  are  equiangular  ;  consequently 
they  are  similar  (p.  18). 


110  GEOMETRY. 

Secondly.  K  the  side  DE  is  per- 
pendicular to  BA,  and  the  side  FD 
to  CA,  the  two  angles  /  and  H 
of  the  quadrilateral  DHAI  are  right 
angles ;  and  since  all  the  four  angles 
are  together  equal  to  four  right 
angles   (b.  i.,  p.  26,  c.  1),   the  remain-     B  G  C 

ing  two  lAH^  IDE,  are  together  equal  to  two  right 
angles.  But  the  sum  of  the  angles  EDF^  IDIT^  is  also  equal 
to  two  right  angles  (b.  l,  p.  1):  hence,  the  angle  FDFis  equal 
to  lAR,  or  BA  C  (a.  3).  In  like  manner,  if  the  third  side  EF 
is  perpendicular  to  the  third  side  BC,  it  may  be  shown 
that  the  angle  BEE  is  equal  to  C,  and  BEE  to  B :  hence, 
the  triangles  ABO,  BEE,  which  have  the  sides  of  the  one 
perpendicular  to  the  corresponding  sides  of  the  other,  are 
equiangular  and  similar  (p.  18). 

Scholium.  In  the  case  of  the  sides  being  parallel,  the 
homologous  sides  are  the  parallel  ones :  in  the  case  of  their 
being  perpendicular,  the  homologous  sides  are  the  perpen- 
dicular ones.  Thus,  in  the  latter  case,  BE  is  homologous 
with  BA,  BE  with  AC,  and  EF  with  BC. 

The  case  of  the  perpendicular  sides  may  present  a  rela- 
tive position  of  the  two  triangles  different  from  that  exhi- 
bited in  the  diagram.  But  we  can  always  conceive  a  tri- 
angle FEB  to  be  constructed  within  the  triangle  ABC,  and 
such  that  its  sides  shall  be  parallel  to  those  of  the  triangle 
compared  with  BAC ]  and  then  the  demonstration  given  in 
the  text  will  apply. 


PEOPOSITIOX  XXII.     THEOREM. 

In  any  triangle,  if  a  line  he  draivn  parallel  fc  the  hose,  xU 
lines  drawn  from  the  vertex  will  divide  the  hose  and  the 
parallel  into  proportional  parts. 

Let   BAG  be   a   triangle,  BE  parallel  to  the  base  BO^ 
and  the  other  lines  drawn  as  in  the  figure;    then 

BI    '.     BE    ::     IK    :    EG    ::     KL    :     GK 


BOOK   lY. 


Ill 


IK 


FG, 


For,  since  DI  is  parallel  to  BF^ 
tlie  triangles  IDA  and  FBA  are 
equiangular ;    and  we  liave 

DI    :     BF    ::    AI    -.    AF ; 

and,    since   IK  is    parallel    to   FG^ 
we  have,  in  like  manner, 

AI    :    AF    ::    IK    :     FG ; 

hence  (b.  ii.,  p.  4,  c),    DI    :     BF 

In  the  same  manner,  we  may  prove  that 

IK    :    FG    ::    KL    :     GH\ 

and  so  with  the  other  segments :  hence,  the  line  DE  is 
divided  at  the  points  I,  K,  L^  in  the  same  proportion,  as 
the  base  BC  is  divided,  at  the  points  F,   G^  II. 

Oor.  Therefore,  if  BG  were  divided  into  equal  parts  at 
the  points  F,  G^  J7,  the  parallel  DF  would  be  divided  also 
into  equal  parts  at  the  points  I,  K,  L. 


PEOPOSiTiON  XXIII.    theoPve:vi. 

In  a  right-angled  triangle^  if  a  ^erj^endiculaT  is  drawn  from 
the  vertex  of  the  right  angle  to  the  hypothenuse. 

1st.  The  triangles  on  each  side  of  the  perpendicular  are  similat 
to  the  given  triangle^  and  to  each  other: 

2d.  Either  side  about  the  right  angle  is  a  mean  proportional 
between  the  hypothenuse  and  the  adjacent  segment: 

3cZ.  The  p)erpendicular  is  a  mean  proportioned  hetiveen  the  seg- 
ments of  the  hypothenuse. 

Let  BAG  be  a  right-angled  triangle,  and  AD  perpen- 
dicular to  the  hypothenuse  BG. 

First.  The  triangles  BAD  and 
BAG  have  the  common  angle  B^ 
the  right  angle  BDA=BAG,  and 
therefore,  the  third  angle  BAD  of 
the  one,  equal  to  the  third  angle 
G.  of  the  other  (b.  i.,  p.  25,  c.  2) : 
hence,  these  two  triangles  are  similar  (p.  18).     In  the  same 


112  GEOMETEY, 

manner  it  may  be  shown  that  the 
triangles  DA  0  and  BA  C  are  simi- 
lar; hence,  the  three  triangles  are 
all  equiangular   and  similar. 

Secondly.  The  triangles  BAD, 
BAC,  being  similar,  their  homolo- 
gous sides  are  proportional.  But  BD  in  the  small  triangle, 
and  BA  in  the  large  one,  are  homologous  sides,  because 
the  J  lie  opposite  the  equal  angles  BAD,  BOA  (p.  18,  s.) ;  the 
hypothenuse  BA  of  the  small  triangle  is  homologous  with  the 
hypothenuse  BG  of  the  large  triangle :  hence,  the  proportion, 

BD    :     BA    ::     BA     :     BC. 
By  the  same  reasoning  we  have 

DC    :    AO    ::     AG    :     BG] 
hence,  each  of  the  sides  AB,  AG,    is   a   mean  proportional 
between  the  hypothenuse  and  the  adjacent  segment. 

Thirdly.   Since  the  triangles  DBA,  DAG,  are  similar,  we 
have,  by  comparing  their  homologous  sides, 

BD    :     AD    ::     AD    :     DG; 
hence,  the  perpendicular  AD  is  a  mean  proportional  between 
the  segments  BD,  DO,  of  the  hj^pothenuse. 

Scholium.   Since  BD    :     AB    :  :     AB    :     BG, 

we  have  (b.  il,  p.  1,  c),     AB^^=<:>  BDxBG. 

For    a    like    reason,  Ad'^=^^DOxBG; 

lheTefoTe,AB'+AG''=^BDxBO+DGxBG=o={BD+DO)X 

BOoBOxBG^^BG'; 

that  is,  the  square  described  on  the  hypothenuse  BG  is  eguivor 
lent  to  the  sum  of  the  squares  described  on  the  two  sides  BA,  A  C. 
Thus,  we  again  arrive  at  this  property  of  the  right-angled 
triangle,  and  by  a  path  very  different  from  that  which  for- 
merly conducted  us  to  it:  and  thus  it  appears  that,  strictly 
speaking,  this  property  is  a  consequence  of  the  more  gen- 
eral property,  that  the  sides  of  equiangular  triangles  are 
proportional.  Thus,  the  fundamental  propositions  of  geom- 
etry are  reduced,  as  it  were,  to  this  single  one,  that  cqui- 
angular  triangles  have  their  homologous  sides  proportional 


BOOK    IV.  113 

It  happens  frequently,  as  in  tliis  instance,  tliat  by 
deducing  consequences  from  one  or  more  propositions,  we 
are  led  back  to  some  proposition  already  proved.  In  fact, 
tlie  chief  characteristic  of  geometrical  theorems,  and  one 
indubitable  proof  of  their  ceilainty  is,  that,  however  we 
combine  them  together,  proviiled  that  our  reasoning  be 
correct,  the  results  v»^e  obtain  alwaj^s  agree  with  each  other. 
The  case  would  be  different,  if  any  proposition  ^vere  false 
or  only  aj)proximatcly  true:  it  would  frequently  happen 
that  on  combining  the  i)ropositions  together,  the  error 
would  increase  and  become  perceptible.  Examples  in  which 
the  conclusions  do  not  agree  with  each  other,  are  to  be  seen 
in  all  the  demonstrations,  in  which  the  redaciio  ad  ahsurdum 
is  employed.  In  such  demonstrations,  if  the  hj^pothesis  is 
untrue,  a  train  of  accurate  reasoning  leads  to  a  manifest 
absurdity  :  that  is,  to  a  conckision  in  contradiction  to  a 
principle  previously  establislied :  and  from  this  we  conclude 
that  the  hypothesis  is  fabe. 

Cor.  If  from  the  point  A^  in  the 
circumference  of  a  circle,  two  chords 
BA^  AC,  be  drawn  to  the  extremi- 
ties of  a  diameter  BC,  the  triangle 
BAC  Avill  be  right-angled  at  A  (b. 
III.,  P.  18,  C.  2) ;  hence,  first,  the  j^erpeiidicular  AD  is  a  mean 
proportional  between  the  two  serjments  BD^  DC,  of  the  dicnneter^ 

hence,  ATj'oBDxDC 

Furthermore,  by  the  proposition,  the  chord  BA  is  a  mean 
proportional  between  the  diameter  BC,  and  the  adjacent  segment 
BD,  that  is, 

bToBCxBD,  and  AO^' ^o=BCxCD. 

PROrOSITION    XXIV.     THEOEEM. 

Two  triangles  having  an  angle  in  each  equal,  are  to  each  other 
as  the  rectangles  of  the  adjacent  sides. 

Let  ABC,  ADE,  be  two  triangles  having  the  equal  angles 
4,  placed,  the  one  on  the  other;    then  the  triangle 
ABC    :     ADE    ::     ABxAG    :     ADkAE, 
8 


114 


GEOMETEY 


Draw  BE.  Then,  the  triangles  ABE, 
ABE,  having  the  common  vertex  E,  and 
their  bases  in  the  same  straight  line,  are  to 
each  other  as  their  bases,  (p.  6,  c.)  that  is 

BAE    :     BAE    ::     BA     :     BA. 

In  like  manner,  since  ^  is  a  common 
vertex,  the  triangle 

BAC    :     BAE    ::    AC    :     AE. 

Multiply  together  the  corresponding  terms  of  these  propor- 
tions, omitting  the  common  factor  BAE]  and  we  have  (b.  ii., 
P.  13), 

BAC    :     BAE    :     BAxAC    :     ABxAE. 

Cor.    If  the  tAYO  triangles  are  equiva- 
lent, we  have, 

BAxAC^^BAxAE: 

hence  (b.  ii.,  p.  2), 

BA     :     BA     :     AE    :     AC: 

consequently,    BC  and   BE  are    parallel     ^ 
(P.  16). 


PEOPOSITION  XXV.     THEOREM. 

Similar  triangles  are   to   each  other  as  the  squares  described  on 
their  homologous  sides. 

Let  ABC,  BEE,  be  two  similar  triangles,  having  the 
angle  A  equal  to  B,  and  the  angle  B=E:  then  will  the 
triangle  BAC  be  to  the  triangle  EBE,  as  a  square  describ- 
ed on  any  side  of  BAC  to  a  square  described  on  the 
homologous  side  of  EBE. 

First,    by  reason   of   the   equal   an- 
gles A  and  B,  we  have  (p.  24), 

BAC  :  BEE  :  :   BAxAC  :   BExBF. 

Also,  because  the  triangles  are  similar 
(p.  18), 

BA    :    BE    ::    AC    :    DF, 


BOOK   IV.  115 

And  multiplj^ing  the  terms  of  this  proportion  bj  the 
corresponding  terms  of  the  identical  proportion 

AC    :     DF    ::     AG    :     DF, 
there  will  result 

BAxAC    :     DExDF    :  :    AC'^    :    Hf"" 
Consequently  (b.  ii.,  P.  4,  c), 

BAG    :     DBF    :  :     AG^    :     BF". 

Therefore,  the  similar  triangles  BAG^  EDF^  are  to  each 
other  as  the  squares  described  on  their  homologous  sides 
AG,  BFy  or  as  the  squares  described  on  any  other  two 
homologous  sides. 

TROPOSITION    XXVI.     TIIEOEEM. 

Two  similar  polygons  may  he  divided  into  the  same  mimher  of 
triangles^  similar  each  to  each,  and  similarly  placed. 

Let  AFBGBj  FKIIIG^  be  two  similar  poh^gons. 

From  the  vertex  of 
any  angle  A^  in  the  poly- 
gon AEDGB,  draw  di- 
agonals, AB,  AG.  From 
the  vertex  of  the  homo- 
logous angle  F,  in  the 
other  polygon,  draw  the 
diagonals    FJ,    FIT,   to  the  vertices  of  the  other  angles. 

The  polygons  being  similar,  the  homologous  angles, 
ABG^  FGIIj  are  equal,  and  the  sides  AB,  BG^  proportional 
to  FG,  Gil,  that  is, 

AB  :  FG  ::  BG  :  Gil  (d.  1). 
Wherefore,  the  triangles  ABG,  FGII,  have  an  angle  in  each 
e(|ual,  and  the  adjacent  sides  proportional:  hence,  they  are 
similar  (p.  20);  consequently,  the  angle  BGA  is  equal  to 
OIIF.  Taking  away  these  equal  angles  from  the  equal 
angles  BGB,  GUI,  and  there  remains  ACD=FIIL  But 
since  the  triangles  ABG,  FGII,  are  similar,  we  hav^e 

AG    '.    FH    ::     BG    :     GIF, 
and  since  the  polygons  are  similar,  '  * 


116 


GEOMETllY. 


BC    :     Gil     ::     CD    :     HI-, 
Lence,  AC    :     FH    :  :     CD    :    HI. 

The  angle  ACD^  we  already  know,  is  equal  to  Fill] 
hence,  the  triangles  ACD,  FIII^  are  similar  (i\20).  In  tke 
game  manner,  it  may  be  shown  that  all  the  remaining  tri- 
angles are  similar,  whatever  be  the  number  of  sides  in  the 
pol^'gons  proposed:  therefore,  two  similar  polygons  may  be 
divided  into  the  same  number  of  triangles,  similar,  and 
similarly  placed. 

Scholium.  The  converse  of  the  proposition  is  equally 
true:  If  two  i^ohjfjons  are  composed  of  the  same  numher  of 
trianrjles  similar  and  similarly  situated,  the  two  joohjgons  are 
similar. 

For,  the  similarity  of  the  respective  triangles  will  give 
the  angles, 

ABC^FGII,  BCA=GIIF,  ACD^FHIi 
hence,         BCD=GIII,  likewise,   CDE^EIK,  &c. 
Moreover,  we  have, 
AB  '.  FG   -.'.   BC  '.   Gn  '.:    CD  :    HI  :  :  DE  :   IK,  &c. ; 

hence,  the  two  polygons  have  their  angles  equal  each  to  each, 
and  their  sides  proj^ortional ;  consequently,  they  are  similar. 


rEOrOSITION  xxvii.    theoeem. 

The  ^;eri?7ie^£'7'S  of  similar  polygons  are  to  each  other  as  their 
homologous  sides :  and  the  j^olygons  are  to  each  other  as  Uie 
squai'es  described  on  tJiese  sides. 

Let    AEDCB   and  FKIIIG,    be   two   similar   polygons: 
then 

per.  AEDCB    :     per.  FKIIIG   :  :   AE 

First.  Since  the  fig- 
ures are  similar,  we 
Lave 

AB  :  FG  ::  BC  : 
GE  ::  CD  :  III,  kc, 
hence,  the  sum  of  the 
antecedents  ABJ-BC+ 


BOOK    lY.  117 

CD^  &c.,  T\-liicli  makes  up  the  perimeter  of  the  first  poly- 
gon, is  to  the  sum  of  the  consequents  FG+GII-VIU^  &;c., 
which  makes  up  the  perimeter  of  the  second  po]3'gon,  aa 
any  one  antecedent  is  to  its  consequent  (i3.  n.,  r.  10) ;  that 
is,  as  AB  to  FG^  or  as  any  other  two  homologous  sides. 

Secondly.  Since  the  triangles  ABC,  FGII,  are  similar, 
we  have  (r.  25), 

ABC    :     FGII    ::     AG'    :    ZZ?'; 
and  from  the  similar  triaiigles  ACD^  Fill, 

ACD    :     Fill    :  :     AC^'    :     FTf  \ 

therefore,  by  reason  of  the  common  ratio,  AC  to  FIF^  we 
have  (b.  ii.,  p.  4,  c.) 

ABC    :     FGII    ::     ACD    :    FUL 

By  the  same  reasoning,  we  should  find 

ACD    :     Fill    :  :     ADE    :     FIK; 

and  so  on,  if  there  were  more  triangles.  And  from  this 
series  of  equal  ratios,  we  conclude  that  the  sum  of  the 
antecedents  ABC-{-ACD-\-ADE,  which  makes  up  the  poly- 
gon AEDCB^  is  to  the  sum  of  the  consequents  FGII+ 
FIII+FIK,  which  makes  up  the  polygon  F/v/IlG,  as  one 
antecedent  ABC^  is  to  its  consequent  FG/l  (n.  ii.,  p.  10),  or 
as  AB"  is  to  FG"  (p.  25);  hence,  similar  jxjhjfjoris  are  to 
each  other  as  the  squares  described  on  their  homolo'joas  sides. 

Cor.  If  iliree  similar  fvjnres  are  descrihed  on  the  tJiree 
sides  of  a  right-angled  triangle,  t/te  figure  on  the  hgijodienuse  in 
equivalent  to  tlie  sum  of  the  oOier  tu:o. 

Let  A,  B,  (7,  denote  three  similar  figures  described  on 
the  hypothenuse  and  sides  of  a  right-angled  triangle,  and  nr, 
6,  c,  the  corresponding  squares;    then, 

A     :     B    :     C    ::     a     :     h     :     c; 

and,,        A     :     B+C    :  :     a    :     ?>-f  c  (b.  ii.,  p.  9)  : 
but,  a  is  equivalent  to  ^  +  c  (p.  11); 

hence,  A  is  equivalent  B  +  C. 


118 


GEOMETRY. 


TEOrOSITION   XXVIII.      TIIEOKEM. 

If  two  chords   intersect  each  other   in  a  circle,  tlie  sejments  are 
reciprocalhj  irroportional. 

Let  tlie  cliords  AB  and  CD  intersect  at  0 :   then 
AO    :     DO     ::     OC    :     OB. 

Draw  AC  and.  BD.  In  the  triangles 
AOC,  DOB,  the  angles  at  0  are  equal, 
being  vertical  angles  (b.  i.,  p.  4)  :  the  angle 
A  is  equal  to  the  angle  D,  because  both  are 
inscribed  in  the  same  segment  (b.  hi.,  p.  18, 
c.  1) ;  for  the  same  reason  the  angle  C=B] 
the  triangles  are  therefore  similar  (p.  IS),  and  the  homologous 
sides  give  the  proportion 

AO    :     DO    ::     CO    :     OB. 

Cor.    Therefore, 

AOxOB=c=DOxCO: 
hence,  the  rectangle  of  the   two   segments  of  one  chord  ls 
equivalent   to    the   rectangle    of    the    two   segments  of  the 
other. 


PROPOSITION  XXIX.     THEOREM. 

If  from  a  j^cint  witliout  a  circle,  tico  secants  he  draicn  termi- 
nating in  tlie  concave  arc,  the  ichole  secants  will  he  recipro- 
cally iiroportional  to  their  external  segments. 

Let  the  secants   OB,    OC,  be  drawn    from   the   point   0: 
then 

OB  :  OC  ::  OD  :  OA. 
For,  drawing  A  C,  BD,  the  triangles 
AOC,  BOD  have  the  angle  0  common  ; 
likewise  the  angle  B=C  (b.  hi.,  p.  18,  c.  1) ; 
these  triangles  are  therefore  similar  (p.  18), 
and  their  homologous  sides  give  the  pro- 
portion, 

OB    :     OC    :  :     OD    :     OA. 
Cor.    Hence,  the  rectangle 

OBxOA^o=OCxOD. 


BOOK    IV. 


119 


Scholium.  This  proposition,  it  may  be  observed,  bears  a 
close  analogy  to  liie  preceding,  and  differs  from  it  only  as 
tlie  two  chords  AB^  CD^  instead  of  intersecting  each  other 
within,  cut  each  other  without  the  circle.  The  following 
proposition  may  be  regarded  as  a  particular  case  of  the 
proposition  just  demonstrated. 


PEOrOSITION  XXX.     THEOEEM. 

If  from  a  point  without  a  circle,  a  tangent  and  a  secant  he 
drawn^  the  tangent  will  he  a  mean  proportional  between  the 
secant  and  its  external  segment. 

From  the  point  0,  let   the   tangent  OA^  and  the  secant 
0(7  be  drawn,  then 

OG    :     OA    ::     OA    :     OD, 
or,  oToOGxOD. 

For,  drawing  AD  and  AC,  the  trian- 
gles DA  0,  CA  0,  have  the  angle  0  com- 
mon ;  also,  the  angle  OAD,  formed  by  a 
tangent  and  a  chord,  is  measured  by  half 
the  arc  AD  (b.  hi.,  p.  21) ;  and  the  an- 
gle G  has  the  same  measure  (b.  hi.,  p.  18); 
hence,  the  angle  OAD=  G  (a.  1)  :  there- 
fore, the  two  triangles  are  similar,  and 
we  have  the  proportion 

OG    :     OA    ::     OA    :     OD, 
which  gives  OZ^  =a=OCx  OD. 


PEOPOSITION  XXXI.     THEOEEM. 


If  either  angle  of  a  triangle  is  bisected  hy  a  line  terminating  in 
tlie  opposite  side^  the  rectangle  of  the  sides  about  the  bisected 
angle,  is  equivalent  to  the  square  of  the  bisecting  line,  together 
with    the    rectangle   contained   by    the   segments  of  the   tJiii'd 


In  the  triangle  BAG,  let  AD  bisect  the  angle  A]   then 


ABxAGoAD^+BDxDG 


120 


GEO^^rETKt  . 


Describe  a  circle  tliroiigli  the  three 
points  A,B,  (7,(u.  iii.,  PR  )15. 13,  s.) ;  prolong 
AD  till  it  meets  the  circumference  in  U, 
and  draw  C£. 

The  triangle  BAD  is  similar  to  the 
triangle  EAC;  for,  by  hypothesis,  the 
Singh  BAD=EAC;  also,  the  angle  B=E, 
since  they  are  both  measured  by  half  the  arc  A  C  (b.  hi.,  p. 
18) ;  hence,  these  triangles  are  similar,  and  the  homolo- 
gous sides  give  the  proi)ortion 

BA     :     AE    ::     AD    :     AC', 
hence,    BAxACoAExAD-,   but  AE=AD+DE, 
and  multiplying  each  of  these  equals  by  AD,  we  have 

AExAD=oAlf+ADxDE', 
now  (p.  28,  c),       ADxDE^=BDxDC] 
hence,  finally,   BAxAC^^AD'-^BDxDC. 


PKOPOSITIOX  XXXII.     TIIEOEEM. 

Tn  any  triangle^  tlte  rectangle  contained  hy  two  sides  is  eqidva- 
lent  to  tlte  rectangle  contained  by  the  diameter  of  tlte  circum- 
scribed  circle^  and  tlte  j^c^'J^cndicular  let  fall  on  the  third  side. 

In  the  triangle  BAC,  let  AD  be  drawn  perpendicular  to 
JbC  ]  and  let  EO  be  the  diameter  of  the  circumscribed 
circle :    then  "will 

ABxACo^ADxCE. 

"For,  drawincr  AE.  the  triano-les 
DBA,  CAE,  are  right-angled,  the  one 
at  D,  the  other  at  A  :  also,  the  angle 
B=E  (b.  III.,  p.  18,  c.  1) ;  these  tri- 
angles are  therefore  similar,  and  we 
have 

AB    :     CE    ::     AD    : 

and  hence,    '  ABxACoCExAD. 

Cor.  If  these  equal  quantities  be  multiplied  by  DC,  there 
^vill  result 

ABxACxBC=CExADxBC; 


AC; 


BOOK    IV 


121 


now,  Al^XBC  \s  double  the  area  of  the  triangle  (P.  6) ; 
therefore,  tlie  iwoduct  of  the  three  sides  of  a  triaiKjIe  U  equal 
to  its  area  multij^lied  by  twice  the  diameter  of  the  circumscribed 
circle. 

The  product  of  three  lines  is  sometimes  represented 
by  a  solid^  for  a  reason  that  will  be  seen  hereafter.  Its 
value  is  easily  conceived,  by  supposing  the  lines  to  be 
reduced  to  numbers,  and  then  multiplying  these  numbers 
together. 

Scholium.  It  may  also  be  demonstrated,  that  tJ/e  area  of 
a  triangle  is  equal  to  its  2'><^rimeler  multlqAied  by  ItaJf  the  radius 
of  the  inscribed  circle. 

For,  the  triangles  AOB, 
BOC,  AGO,  which  have  a 
common  vertex  at  0,  have 
for  their  common  altitude  the 
radius  of  the  inscribed  circle ; 
hence,  the  sum  of  these  tri- 
angles will  be  equal  to  the 
sum  of  the  bases  xlZ>,  BC^  A  (7,  multiplied  by  half  the 
radius  0D\  hence,  the  area  of  the  triangle  ABC  is  eqiicd 
pj  its  perimeter  muUiqAied  by  half  the  radius  of  the  inscribed 
circle. 


rRorosiTioN  xxxiii.    tiieoeem. 

In  every  quxtdrilateral  inscribed  in  a  circle,  tJie  rectanyle  of  the 
two  diayoiia/s  is  equiualent  to  the  sum  of  the  reciawjlcs  of 
the  oqrposite  sides. 

Let  A  BCD  be  a  quadrilateral  inscribed  in  a  circle,  and 
AC,  BD,  its  diagonals:    then  we  shall  have 

ACxBD<:>=ABxCD+ADxBG. 

Take  the  arc  CO=uiD,  and  draw 
BO,  meeting  the  diagonal  AC  in  /. 

The  angle  ABD=CBI,  since  the 
one  has  for  its  measure  half  of  the  arc 
AD  (b.  II r.,  p.  18),  and  the  other,  half 
of  CO,  equal  to  Al)\  the  angle  jVJJB 
~  BCI,  because  they  are  subtended  by 


122 


GEOMETRY, 


the  same  arc  ;  hence,  the  triangle 
ABD  is  similar  to  the  triangle  IBG^ 
and  Ave  have  the  proportion 

AD    :     CI    ::     BD    :     BC  \ 
and  consequently, 

ADX  BC<z=  CIX  BD.  ^ 'O 

Again,  the  triangle  ABI  is  similar  to  the  triangle  BDC] 
for  the  arc  AD  being  equal  to  CO,  if  OD  be  added  to 
each  of  them,  Ave  shall  have  the  arc  AO=DC;  hence,  the 
angle  ABI  is  equal  to  DBC;  also,  the  angle  i?.-!/ to  BDC^ 
because  they  stand  on  the  same  arc;  hence,  the  triangles 
ABI,  DBC,  ai'e  similar,  and  the  homologous  sides  give  the 
proportion 

AB    :     BD     :  :     AI    :     CD; 
hence,  ABxCD=c^AIX  BD. 

Adding  the  two  results  obtained,  and  observing  that 
AIxBD+CIxBD={AI+CI)xBD=ACxBD, 
we  shall  have 

ADxBC+ABxCD=c^ACxBD. 


PEOBLEMS 
RELATING   TO   THE    FOUETH    BOOK. 


pkoblem:  I. 

To  divide  a  given  straight  line  into  any  nuniber  of  equal  partSf 
or  into  parts  proportiorial  to  given  lines. 

First.  Let  it  be  proposed  to  divide  the 
line  AB  into  five  equal  parts.  Through 
the  extremity  A,  dra^v  the  indefinite  straight 
line  A  G :  take  li  C  of  any  mag-nitude,  and 
apply  it  five  times  upon  A  G  ;  join  the  last 
point  of  division  G,  and  the  extremity  B 
of  the  given  line,  by  the  straight  line  GB'^ 
then  through   C,  draAV  CI  parallel  to   GB\ 


A 

9 

h 

/ 

\ 

V 

|m 

cj 

B 

1 

r 

BOOK    lY. 


123 


Al  will  be  the  fiftli  part  of  the  line  AB ;  and  by  apply- 
ing AI  five  times  npon  AB^  the  line  AB  will  be  divided 
into  five  equal  parts. 

For,  since  CI  is  parallel  to  GB^  the  sides  AG^  AB^ 
are  cut  proportionally  in  G  and  /  (p.  15).  But  AG  \^ 
the  fifth  part  of  AG,  hence,  AI  is  the  fifth  part  of  AB, 

Secondly.  Let  it  be 
proposed  to  divide  the 
line  AB  into  parts  pro- 
portional to  the  given 
lines  P,  Q,  R.  Through 
A,  draw  the  indefinite 
Une  A G  ;    make  AO=P, 

extremities  E  and  B ;  and  through  the  points  C  and  D, 
draw  67,  BF,  parallel  to  EB ;  the  line  AB  will  be  divided 
into  parts  AI,  IF,  FB,  proportional  to  the  given  lines  P,  Q,  R. 
For,  by  reason  of  the  parallels  CI,  BE,  EB,  the  parts 
AI,  IF,  EB,  are  proportional  to  the  parts  A  C,  CD,  BE  (p. 
15,  c.  2) ;  and  by  construction,  these  are  equal  to  the  given 
lines  P,  Q,  R. 


PROBLEM  II. 


To  find  t^  fijurth  proportional  to  three  given  lines,  A,  B,  C, 

Draw  the  two  indefi- 
nite lines  BE,  BE,  form- 
ing any  angle  with  each 
other.  Upon  BE  take 
BA^A,  and  BB=B; 
upon  BE  take  BC=G, 
draw  AC;  ana  through 
the  point  B,  draw  BX  parallel  to  AC',  and  DX  will  be 
the  fourth  proportional  required.  For,  since  BX  is  parallel 
to  ACy  we  have  the  proportion  (p,  15,  C  1), 

BA     :     BB    ::     DC    :     BX; 
now,  the  first  three   terms  of  this  proportion  are  equal   to 
the  three  given  lines :   consequently,  BX  is  the  fourth  pro- 
portional required. 


124: 


GEOMETEY. 


Cor.  A  tliird  proportional  to  two  given  lines,  A,  B,  may 
be  found  in  the  same  manner,  for  it  will  be  the  same  as  a 
fourth  proportional  to  the  three  lines,  J,  i?,  B. 


teoblem:  III. 

To  find  a  mean  proportional  heticeen  iico  given  lines  A  and  B, 

Upon  the  indefinite  line  DF^ 
take  J)E^A,  and  EF=B\  and 
upon  the  whole  line  BF^  as  a 
diameter,  describe  the  semicircum- 
ference  BGF  \  at  the  point  E^ 
erect,  upon  the  diameter,  the  per- 
pendicular EG  meeting  the  semicircumference  in  G\  EG 
will  be  the  mean  proportional  required. 

For,  the  perpendicular  EG^  let  fall  from  a  point  in  the 
circumfereix^e  upon  the  diameter,  is  a  mean  proportional 
between  the  two  segments  of  the  diameter  BE^  EF  (p.  28, 
C.) ;  and  these  segments  are  equal  to  the  given  lines  A  and  B. 


TKOBLEM  IV. 

To  divide  a  given  line  into  two  such  parts,  thai  the  greater 
part  shall  be  a  mean  proportional  between  the  whole  line 
and  the  other  p>art. 

Let  yi^  be  the  given  line. 

At  the  extremity  B,  erect  the 
perpendicular  BC^  equal  to  the 
lialf  of  AB ;  from  the  point  C, 
as  a  centre,  with  the  radius  CB, 

describe  a  semicircle  ;    draw  AC 

cutting  the  circumference  m  B\     ^  ^         ^^ 

and  take  AF=AB'.    then  F  will  be  the  point  of  division, 

and  we  shall  have, 

AB    :     AF    ::     AF    :     FB. 

For,   AB  being  perpendicular   to   the   radius   at   its  ex- 
tremity, is  a  tangent  (b.  III.,  P.  9) ;    and  if  AC  be  prolonged 


BOOK    TV. 


125 


till  it  again  meets  tlie  circumference,  in  E^  we  shall  have 
(p.  SO), 

AE    :    AB    ::     AB    :    AD', 

hence,  bj  division, 

AE-AB    :    AB    ::    AB-AD    :    AD. 

But,  since  the  radius  is  the  half  of  AB,  the  diameter  DE 
is  equal  to  AB,  and  consequently,  AE-~AB=AIJ=AF  ] 
also,  because  AF=AB,  we  have  AB—AB=FB:  hence, 

AF    :     AB    ::     FB    :    AD^ovAF] 

whence,  bj  inversion, 

AB    :     AF    ::     AF    :     FB. 

Scholium.  This  sort  of  division  of  the  line  AB,  viz.,  so 
that  the  ivltole  line  sliall  he  to  the  (jreater  j^cirt  as  the  greater  ixirt 
is  to  the  less,  is  called  division  in  extreme  and  mean  ratio. 
It  may  further  be  observed,  tlint  the  secant  AE  is  divided 
in  extreme  and  mean  ratio  at  the  jooint  B',  for,  since  AB- 
LE, we  have, 

AE    :    BE    '.'.    DE    :    AD. 


TEOBLEM  V. 

Tlirough  a  given  point,  in  a  given  angle,  to  draw  a  line  so 
that  the  segments  compreliended  between  the  2)oint  and  tlie 
two  sides  of  tlie  angle,  shall  he  equal. 


Let  BCD  be  the  given  angle,  and  A  the  given  point. 

Through  the  point  A,  draw  AE 
parallel  to  CD,  make  BE^CE,  and 
through  the  points  B  and  A,  draw 
BAD  ;    this  will  be  the  line  required. 

For,  AE  being  parallel  to  CD,  we 
have, 

BE    '.     EG    '.'.    BA    :    AB-, 
but  BE=EG',   therefore,  BA=AD. 


126 


GEOMETRY. 


PEOBLEM  n. 


To  describe  a  sjiiare  that  shall  he  equivalent  to  a  given  parol- 
lelogram,  or  to  a  given  triangle. 


D 


A 


First  Let  ABCD  be 
tlie  given  parallelogram, 
AB  its  base,  and  LE 
its  altitude:  between  AB 
and  DE  find  a  mean 
proportional  A'F;  then 
will  the  square  described  upon  XY  be  equivalent  to  the 
parallelogram  ABCD. 

For,  by  construction, 

AB    :     XY    :  :     XY    :     DE ; 
therefore,  XY^^  o=ABxDE', 

but  ABxDE  is  the  measure  of  the  parallelogram  (p.  5), 
and  XY'  that  of  the  square;  consequently,  they  are  equiv- 
alent. 

Secondly.  Let  BAC 
be  the  given  triangle, 
EC  its  base,  AD  its  al- 
titude :  find  a  mean  pro- 
portional between  BG 
and  the  half  of  AD,  and  I^  ^     ^ 

let  XY  be  that  mean  ;  the  square  described  upon  XY 
will  be  equivalent  to  the  triangle  ABC. 

For,  since 

BG    :     XY    ::     XY    :     \AD, 
it  follows,  that 

Xf'^o^BCxlAD', 
hence,  the  square  described  upon  XY  is  equivalent  to  the 
triangle  BAC. 


PROBLEM   VII. 


Upon  a  given  line,  to  describe  a  rectangle   that  shall  he  equiva* 
lent  to  a  given  rectangle. 


Let  AD  be  the  line,  and  ABFC  the  given  rectangle. 


BOOK    lY.  127 


Find   a    fourth    pro-  X 

portional    to    the    three      Ci iF 

lines,  AD^  AB,  AC,  and 
let  AX  be    that   fourth 


proportional  ;    a   rectan-      ^  -^  A  D 

gle  constructed  with  the 

sides  AD  and  AX  will  be  equivalent  to  the  rectangle  ABFC. 
For,  since 

AD    :     AB    ::     AC    :     AX, 
it  follows,  that       ADxAXoABxAG', 

hence,  the  rectangle  AD  EX  is  equivalent  to  the  rectangle 
ABFC. 

PEOBLEM  YTil. 

To  find  two  lines  ivliose  ratio  shall  he  the  same  as  the  ratio  of 
two  rectangles  contained  hy  given  lines. 

Let  AxB^   CxD^  be    the    rectangles    contained    bj  tho 
given  lines  A,  B,  C,  and  D. 

Find  X,  a  fourth   proportional   to   the 

three   lines,    B,   C,  D;    then  will   the   two  :^'  ~"     ' 

lines  A   and   X  have    the    same    ratio    to  ' 

each   other    as    the    rectangles  AxB  and  -p.^ ^ 

CXB.  X. . 


For  since, 

B    :     G    ::     D    :     X, 

it  follows  that  CxDoBxX]    hence, 

AxB    :     CxD    ::     AxB    :     BxX    :  :     A    :     X. 

Cor.  Hence,  to  obtain  the  ratio  of  the  squares  described 
upon  the  given  lines  A  and  (7,  find  a  third  proportional  X^ 
to  the  lines  A  and  (7,  so  that 


A    :     C    ',',     G    :    X 


you  will  then  have 


^xXo(7',  or  ^'xXo^X(7^   hence, 
a:    :     ^'    ::     A     :     X 


128  GEOMETRY. 

niOBLEM   IX. 
To  find  a  triangle  that  sliall  he  equivalent  to  a  given  yAggoji, 

Let  A  ED  CD  be  tlic  given  polygon. 

First.  Draw  the  diagonal  CD 
cutting  olF  the  triangle  CDE; 
througli  the  point  D,  draw  DF 
parallel  to  CD,  meeting  AD  pro- 
longed, in  F;  draw  CF:  the  poly- 
gon AD  I)  CD  is  equivalent  to 
the  polygon  AFCB,  Aviiicli  has  one  side  less  than  the 
given  polygon.  - 

For  the  triangles  CDF,  CFF,  have  the  base  CF  com- 
mon, they  have  also  equal  altitudes,  since  their  vertices  B 
and  F,  are  situated  in  a  line  DF  parallel  to  the  base: 
these  triangles  are  therefore  equivalent  (p.  2,  c.)  Add  to 
each  of  them  the  figure  AFCD,  and  there  will  result  the 
polygon  ADDCB^  equivalent  to  the  polygon  AFCB. 

The  angle  B  may  in  like  manner  be  cut  off",  by  sub- 
stituting for  the  triangle  ABC,  the  equivalent  triangle  J.  (7 C, 
and  thus  the  pentagon  AFDCB  will  be  changed  into  an 
equivalent  triangle   G  CF. 

The  same  process  may  be  applied  to  every  other  figure; 
for.  by  successively  diminishing  the  number  of  its  sides, 
one  being  retrenched  at  each  step  of  the  process,  the  equiv- 
alent triangle  will  at  last  be  found. 

Scholium,  AYe  have  already  seen  that  every  triangle  may 
be  changed  into  an  equivalent  square  (pkob.  6);  and  thus 
a  square  may  alwaj^s  be  found  equivalent  to  a  given  recti- 
lineal iigui'e,  which  operation  is  called  squaring  the  recti 
liujal  figure,  or  the  quadrature  of  it. 

The  problem  of  tlie  quadrature  of  tlie  circle  consists  in 
finding  a  square  equivalent  to  a  circle  whose  diameter  is 
given. 


BOOK    lY.  129 


PKOBLEM    X. 

To  find  the  side  of  a  square    which   shall   be   equivalent   t/j   Hie 
sum  or  the  difference  of  two  given  squares. 

Let  A  and  B  be  the  sides  of  the  given  squares. 

First.   If  it  is  required  to  find  -p 

a  square  equivalent  to  the  sum 
of  these  squares,  draw  the  two 
mdefniite  hnes,  ED^  EF,  at  right 
angles  to  each  other ;  take  ED= 
A,  and  EG=B]  and  draw  I)G: 
this  will  be  the  required  side  of  the  square. 

For  the  triangle  BEG  being  right-angled,  the  square 
described  upon  the  hypothenuse  BG^  is  equivalent  to  the 
sum  of  the  squares  upon  EB  and  EG   (p.  11). 

Second! y.  If  it  is  required  to  find  a  square  equivalent 
to  the  difference  of  the  given  squares,  form,  as  before, 
the  right  angle  FEU]  take  GE  equal  to  the  shorter 
of  the  sides  A  and  B\  from  the  point  6^  as  a  centre,  with 
a  radius  GII^  equal  to  the  other  side,  describe  an  arc 
cutting  EII  in  //:  the  square  described  upon  EH  will  be 
equivalent  to  the  difference  of  the  squares  described  upon 
the  lines  A  and  B. 

For,  the  triangle  GEII  is  right-angled,  the  hypothenuse 
GII=A,  and  the  side  GE=B ;  hence,  the  square  described 
upon  EII,  is  equivalent  to  the  difference  of  the  squares  A 
and  B  (p.  11,  c.  1). 

Scholium.  A  square  may  thus  be  found,  equivalent  to  the 
sum  of  any  number  of  squares ;  for  a  construction  similar 
to  that  which  reduces  two  of  them  to  one,  will  reduce 
three  of  them  to  two,  and  these  two  to  one,  and  so  of 
others.  It  Avould  be  the  same,  if  any  of  the  squares  were 
to  be  subtracted  from  the  sum  of  the  others. 


130  GEOMETRY. 


PROBLEM   XT. 

Tc  find  a  s(puare  ivhich  shall  be  to  a  given  square  as  one  give)\ 
line  is  to  another  given  line. 

Let  A  C  bo  the  given   square,  and  M  and  N  the  given 
lines. 

Upon    the    indefi-        p    C 

nite  line  EG^  take  EF 
=3f,  and  FG=N\  up- 
on EG  as  a  diameter 
describe  a  semicircum- 
ference,  and  at  the 
point  F  erect  the  per- 


B 


pendicular  FII.  From  the  point  II,  draw  the  chords  HG^ 
HE,  which  produce  indefinitely :  upon  the  first,  take  HE 
equal  to  the  side  AB  of  the  given  squrare,  and  through  the 
point  K  draw  EI  parallel  to  EG ;  HI  will  be  the  side  of 
the  required  square. 

For,  by  reason  of  the  parallels  EI,    GE,  we  have 

HI    :     HE    ::     HE    :     HG ; 

hence,  S7'     :     HE''    :  :     HE''    :     HG''  : 

but  in  the  right-angled  triangle  GHE,  the  square  of  HE 
is  to  the  square  of  HG  as  the  segment  EF  is  to  the  seg- 
ment EG  (p.  11,  c.  3),  or  as  J/  is  to  iV;   hence, 

iTT"     :     he""     :  :     21    :     K 

But  IIE=AB ;  therefore,  the  square  described  upon  HI  is 
to  the  square  described  upon  AB  as  21  is  to  N. 


PEOBLEM    XII. 

Upon   a   given  line,  to  describe   a  ^^olggon  similar  to  a  given 

polygon. 

Let    FG   be   the    given   line,    and    AEDCB    the   given 
polygon. 


BOOK    lY. 


181 


In  the  given  poly- 
gon, draw  the  diago- 
nals A  C,  AD ;  at  the 
point  F  make  the  angle 
GFII^BA  C,  and  at  the 
point  G^  the  angle  FGH 
=ABO',    the  lines  i^i^ 

Gil  will  intersect  each  other  in  II,  and  the  triangle  FGH 
will  be  similar  to  AUG  (p.  18).  In  the  same  manner  upon 
FII,  homologous  to  A  (7,  describe  the  triangle  Fill  similar 
to  ADC',  and  upon  FI,  homologous  to  AD,  describe  the 
triangle  FIK  similar  to  ADE.  The  polygon  FGIIIK  will 
be  similar  to  ABCDE,  as  required. 

For,    these    two    polygons    are   composed  of  the   same 
number  of  similar  triangles,  similarly  placed  (p.  26,  s.) 


PROBLEM   XIII. 

Two  similar  figures   being  given,    to  describe   a  similar  figure 
which  shall  be  equivalent  to  their  sum  or  difference. 

Let  A  and  B  be  homologous  sides  of  the  given  figures. 

Find  a  square  equivalent 
to  the  sum  or  difference  of 
the  squares  described  upon  A 
and  B ;  let  X  be  the  side  of 
that  square ;  then  will  X  be 
that  side  in  the  figure  required, 
which  is  homologous  to  the 
sides  A  and  B  in  the  given  figures.  Let  the  figure  itself, 
then,  be  constructed  on  the  side  X,  as  in  the  last  problem. 
This  figure  will  be  equivalent  to  the  sum  or  difference  of 
the  figures  described  on  A  and  B  (p.  27,  c.) 


PROBLEM    XIV. 


To  desa-ibe  a  fif/ure  similar   to   a  given  figure,  and  bearing  m 
it  the  given  ratio  of  31  to  X. 

Let  A  be  a  side  of  the  given  figure,  X  the  homologous 
side  of  the  required  figure. 


132 


GEOMETRY. 


Find  tlie  value  of  X,  sucli,  that  its 
equare  shall  be  to  the  square  of  ^i,  as  M  to 
N  (PHOB.  11).  Then  upon  X  describe  a  fig- 
ure similar  to  the  given  figure  (pnoB.  12) : 
this  will  be  the  figure  required. 


PEOBLEM   XV. 

To  construct  a  figure  similar  to  tlte  figure  P,  and  equivalent  to 
the  figure   Q. 

Find  M,  the  side  of  a 
square  equivalent  to  the  fig- 
ure P,  and  iV  the  side  of  a 
square  equivalent  to  the  figure 
Q  (PROB.  9,  s.)  Let  X  be  a 
fourth    proportional     to     the 

three  given  lines,  31,  i^^  AB ;  upon  the  side  X,  homologous 
to  AB^  describe  a  figure  similar  to  the  figure  P ;  it  Avill  also 
be  equivalent  to  the  figure   Q. 

For,  calhng   Y  the  figure    described   upon   the   side  X, 
we  have, 

P    :     Y    ::    AK    :     X' ; 
but  by  construction, 

AB  :  X  ::  M  :  N,  or,  Ajf  :  X'  :  :  J/'  :  N^ ', 
hence,  P    :      F    :  :     j/     :     Xl 

But,  by  construction  also, 

lfo=P,  and     X^=o=g- 
therefore,  P    :      Y    ::     P    :     § ; 

consequently,  F=o=  Q  ;  hence,  the  figure  Y  is  similar  to 
the  figure  P,  and  equivalent  to  the  figure   Q. 


PROBLEM     XYl. 


To  construct  a  rectangle  equivalent  to  a  given  square^  and  hav- 
ing Hie  sum  of  its  adjacent  sides  equal  to  a  given  line. 

Let   C  be  the   square,    and   the   line   AB  equal  to  the 
sum  of  the  sides  of  the  required  rectangle. 


BOOK    lY. 


laa 


Wi 


FB 


Upon  AB  as  a  diam- 
eter, describe  a  semicir- 
cumference ;  at  J.,  draw 
AD  perpendicular  to  AB, 
and  make  it  equal  to  the 
side  of  tlie  square  (7; 
then  dr^aw  the  line  i>^ parallel  to  the  diameter  AB;  from  the 
point  F,  where  the  parallel  cuts  the  circumference,  draw 
FF  perpendicular  to  the  diameter ;  AF  and  FB  will  be 
the  sides  of  the  required  rectangle. 

For,  their  sum  is  equal  to  AB ;  and  their  rectangle 
AFxFB  is  equivalent  ta  the  square  of  FF,  or  to  the  square 
of  ^Z> ;  hence,  this  rectangle  is  equivalent  to  the  given 
square   C, 

Scholium.  The  problem  is  impossible,  if  the  distance  AB 
exceeds  the  radius ;  that  is,  the  side  of  the  square  C  must 
not  exceed  half  the  line  AB. 


PROBLEM    XVII. 

To  construct  a  rectangle  that  sJiall  he  equivalent  to  a  ijiven 
square,  and  tJte  difference  of  ivhose  adjacent  sides  shall  he 
equal  to  a  given  line. 

Let  C  denote  the  given  square,  and  AB  the  difference 
of  the  sides  of  the  rectangle. 

Upon  the  given  line  AB,  as  a 
diameter,  describe  a  circumference. 
At  the  extremity  of  the  diameter, 
draw  the  tangent  AD,  and  make  it 
equal  to  the  side  of  the  square  C; 
through  the  point  D  and  the  cen- 
tre 0  draw  the  secant  DOF,  inter- 
secting the  circumference  in  F  and 
F;  then  will  DF  and  DF  be  the 
adjacent  sides  of  the  required  rectangle. 

For,  the  difference  of  these  lines  is  equal  to  tlie  diame 
ter  FF  or  AB;  and  the  rectangle  DF,  DF  is  eciuivalent  tc 
AD^'{p.SO)',  hence,  the  rectangle  i>/^x/^^,  is  equi-aleut 
to  the  given  square  C 


134 


GEOMETKY 


PKOBLEM    XVIII. 

To  find  the  common  measure^  between  tJie  side  and  diagonal  of 

a  square. 

Let  ABCG  be  any  square,  and  AC  its  diagonal. 

We  first  applv  CB  upon  CA. 
For  this  purpose  let  the  semicir- 
cuniference  DBE  be  described,  from 
the  centre  (\  with  the  radius  CB^ 
and  produce  AC  io  E.  It  is  evident 
that  GB  IS  contained  once  in  AC, 
with  the  remainder  AD.  The  result 
of  the  first  operation  is,  therefore,  a 
quotient  I,  with  the  remainder ^4 Z^. 

This  remainder  must  now  be  compared  with  BC^  or  its  equal 
AB. 

Since  me  angle  ABC  is  a  right  angle,  AB  is  a  tangent, 
and  ^mce  AE  is  a  secant  drawn  from  the  same  point,  we 
nav'f'  [V  30), 

AD  :  AB  ::  AB  :  AE. 
tlenoe  m  the  second  operation,  where  AD  is  compared 
with  AB,  the  equal  ratio  of  AB  to  AE  may  be  taken  instead  : 
l)ut  AB,  or  its  equal  CD,  is  contained  twice  in  AE,  Avitli 
the  remainder  AD ;  the  result  of  the  second  operation  is 
therefore  a  quotient  2  with  the  remainder  AD,  and  this  must 
U'  again  compared  with  AB. 

Thus,  the  third  operation  consists  in  comparing  again 
A  D  with  AB,  and  may  be  reduced  in  the  same  manner  to 
the  comparison  of  AB  or  its  equal  CD  with  AE ;  from 
which  there  will  again  be  obtained  a  quotient  2,  and  the 
remainder  AD. 

Hence,  it  is  evident  that  the  process  will  never  termi- 
nate, and  consequently  that  no  remainder  is  contained  in 
its  divisor  an  exact  number  of  times  ;  therefore,  there  is 
no  common  measure  between  the  side  and  the  diagonal  of 
a  square.     This  property  has  already  been  shown,  since  (p. 

11,  c.  6\ 

AB    :     AC    ::     1     :      V2, 
but  it  acquires   a   greater   degree   of   clearness   by  the  geo- 
metrical investigation. 


BOOK    V. 

REGULAR  POLYGONS— MEASUREMENT  OF  THE  CIRCLR 
DEFINITION. 

A  Eegular  Polygon  is  one  wliich  is  botli  equilateral 
and  eauionf^ular. 

A  regular  polygon  may  have  any  nnmlDer  of  sides. 
The  equilateral  triaDgle  is  one  of  three  sides ;  the  square, 
is  one  of  four. 

PROPOSITION  I.     THEOREM. 

Regular  polygons  of  the  same  numhcr  of  sides  are  similar  figures. 

Let  xiBCDEF^  ahcedf  be  two  such  polygons. 
Then,     either     angle,  E D 


as  A^  of  the  polygon 
ABCDEF,  is  equal  to 
twice  as  many  right  an- 
gles less  four,  as  the  fig- 
ure has  sides,  divided  by  A  B 
the  number  of  sides ;  and  the  same  is  true  of  either  angle 
of  the  other  polygon  (b.  i.,  p.  26,  c.  4) ;  hence  (a.  1),*  the 
angles  of  the  polygons  are  equal. 

Again,  since  the  polygons  are  regular,  the  sides  AB^  BC, 
CD^  kc,  are  equal,  and  so  likewise  the  sides  ab^  bc^  cd 
(d.),  &c.  ;   hence 

AB  :   ab   ::   BC  :    be   ::    on  :   cd,  &c.; 

therefore,  the  two  polygons  have  their  angles  equal,  and 
their  sides  taken  in  the  same  order  proportional  ;  conse- 
quently,   they    are  similar  (b.  iv.,  D.  1). 

Cor.  1.  The  perimeters  of  two  regular  polj^gons  of  the 
same  number  of  sides,  are  to  each  other  as  their  homolo- 
gous sides,  and  their  surfaces  are  to  each  other  as  the 
squares  of  those  sides  (b.  IV.,  P.  27). 


136 


GEOMETRY. 


Cor.  2.  The  angle  of  a  regular  polj^gon,  like  the  angle 
of  an  equiangular  polygon,  is  determined  bj  the  number 
of  its  sides  (b.  i.,  p.  26,  c.  4). 

PEOPOSITION   II.     THEOREM. 

4  regular  polygon    may    he   circumscribed  by  the  circumferenca 
of  a  circle,  and  a  circle  may  be  inscribed  luitltin  it 

Let  IIGFE^  &c.,  be  any  regular  polygon. 

Through  the  three  points  A^  B^  (7, 
describe  the  circumference  of  a  circle : 
the  centre  0  will  lie  in  the  line  OP, 
drawn  perpendicular  to  BC  at  the 
middle  point  P  (b.  iil,  p.  6,  s.)  Then 
draw   OB  and   OC. 

If  the  quadrilateral  OPCD  be 
placed  upon  the  quadrilateral  OPBA, 
they  will  coincide ;  for,  the  side  OP  is  common ;  the  angle 
OPC=OPB,  each  beinor  a  rio-ht  ande ;  hence,  the  side 
PC  will  apply  to  its  equal  PB,  and  the  point  C  will 
fall  on  B :  besides,  the  polygon  being  regular,  the  angle 
PCB=PBA  (d.)  ;  hence,  CD  will  take  the  direction  BA ; 
and  since  CD=BA^  the  point  D  will  fall  on  ^1,  and  the 
two  quadrilaterals  will  coincide.  Hence,  OD  is  equal  to 
A  0 ;  and  consequently,  the  circumference  which  -passes 
through  the  three  points  A^  B^  (7,  will  also  pass  throagh 
:he  point  D.  In  the  same  manner  it  may  be  shown,  that 
the  circumference  which  passes  through  the  .  three  points 
5,  C,  P,  will  also  pass  through  the  point  E'^  and  so  of  all 
*;he  other  vertices;  hence,  the  circumference  which  passes 
uhrough  the  points  J.,  B,  C^  passes  also  through  the  vertices 
of  all  the  angles  of  the  polygon,  consequenth',  the  circum- 
ference of  the  circle  circumscribes  the  polygon  (b.  iil,  d.  7). 

Again,  in  reference  to  this  circle,  all  the  sides  AB^  BG^ 
GD^  &c.,  of  the  poh'gon,  are  equal  chords ;  they  are  there- 
fore equally  distant  from  the  centre  (b.  hi.,  p.  8) :  hencft,  if 
from  the  point  (9  as  a  centre,  with  the  distance  OP,  a  cir- 
cumference be  described,  it  Avill  touch  the  side  BC^  and  all 
the  other  sides  of  the  polygon,  each  in  its  middle  point,  and 
the  circle  will  be  inscribed  in  the  polygon  (b.  hi.,  d.  11). 


BOOK    Y. 


187 


Sclwlium.  The  point  (9,  the  common  centre  of  the  in- 
scribed and  circumscribed  circles,  may  also  be  regarded  as 
the  centre  of  the  polygon;  and  the  angle  AOB  is  called 
the  angle  at  the  centre^  being  formed  by  two  lines  drawn 
from  the  centre  to  the  extremities  of  the  same  side  AB. 
The  perpendicular   OP^  is  called  the  apotliem  of  the  polygon 

Cor.  1.  Since  all  the  chords  AB^  BC^  Cl)^  &c.,  are  equal, 
all  the  angles  at  the  centre  are  likewise  equal  (b.  ill.,  P.  4);  and 
therefore,  the  value  of  anj^  angle  will  be  found  by  dividing 
four  right  angles  by  the  number   of  sides   of   the  polygon. 

Cor.  2.  To  inscribe  a  regular 
polygon  of  any  number  of  sides 
in  a  given  circle,  we  have  only 
to  divide  the  circumference  into 
as  many  equal  parts  as  the  poly- 
gon has  sides ;  for,  when  the  arcs 
are  equal,  the  chords  AB,  BC^  CD^ 
kc,  are  also  equal  (b.  hi.,  p.  4) ; 
hence,  likewise  the  triangles  AOB,  BOC,  COD,  must  be 
equal,  because  their  sides  are  equal  each  to  each  (b.  i.,  p.  10) ; 
therefore,  by  addition,  all  the  angles  A  BC,  BCD,  CDE,  &c.,  are 
equal  (a.  2) ;  hence,  the  figure  ABCDEII,  is  a  regular  polygon. 


PROPOSITION  III.     PROBLEM. 
To  inscribe  a  square  in  a  given  circle. 

Draw  two  diameters  AC,  BD,  intersecting  each  other 
at  right  angles;  join  their  extremities  A,  B,  C,  D ^  tho 
figure  A  BCD  will  be  a  square. 

For,  the  angles  AOB,  BOC  kc, 
being  equal,  the  chords  AB,  BC, 
&c.,  are  also  equal  (b.  hi.,  p.  4) : 
and  the  angles  ABC,  BCD,  kc, 
being  inscribed  in  semicircles,  are 
right  angles  (b.  hi.,  p.  18,  c.  2). 

Scholium..  Since  the  triangle 
BCO  is  right-angled  and  isosceles, 

we  have  (b.  iv^  p.  11,  c.  5),  BC  :  BO  -. -.  ^J1  \  I ; 
hence,  the  side  of  the  inscribed  square  is  to  the  radius^  as  the 
square  root  of  two,  to  unit?/. 


138 


GEOMETEY. 


PROPOSITION   IV.     THEOREM. 

If  c  regular  hexagon   he   inscribed   in   a   circle^  its  side  will  he 
equal  to  the  radius. 

Let  ABCDEH^  be   a  regular   hexagon,    inscribed    in    a 
circle:    then  will  its  side  AB  be  equal  to  the  radius  OA. 

For,  the  angle  A  OB  is  equal 
to  one-sixth  of  four  right  angles, 
(p.  2,  e.  1),  or  one-third  of  two 
right  angles  :  hence,  the  sum 
of  the  remaining  angles  OAB^ 
OBA^  is  equal  to  two-thirds  of 
two  right  angles  (b.  i.,  p.  25). 
But  the  triangle  AOB  is  isos- 
celes, hence,  the  angles  at   the 

base  are  equal  (b.  I.,  P.  11) :  therefore  each  is  one-third  ol 
two  right  angles :  hence,  the  triangle  A  OB  is  equiangu- 
kr:    hence,  AB=AO  (b.l,  p.  12). 


PROPOSiTiox  V.    proble:^l 
To  inscribe  in  a  given  circle^  a   regular  hexagon. 

Let  0  be  the   centre,  and  OB  the   radius  of  the  given 
circle. 

Beginning  at  any  point,  as 
B,  apply  the  radius  BO,  six 
times  as  a  chord  to  the  circum- 
ference, and  we  shall  form  the 
regular  hexagon  BCDEFA  (p. 
4).  Ilence,  to  inscribe  a  regu- 
lar hexagon  in  a  given  circle, 
the  radius  must  be  applied  six 
times  as  a  chord,  to  the  cir- 
cumference; which  will  bring 
us  round  to  the  point  of  begin- 


Cor.  1     If  tlie  vertices  of  the  alternate  angles  be  joined 


BOOK   y 


139 


bj  tlie  lines  AC^  OE,  EA^  there  will  be  inscribed  in  the 
circle  an  equilateral  triangle  ACE,  since  each  of  its  angles 
will  be  measured  by  one-sixth  of  four  right  angles,  or  one- 
third  of  two  (b.  l,  p.  25,  c.  5). 

Cor.  2.   If  we  draw  the  radii  OA^  OC,  the  figure  OCBA 
will  be  a  rhombus  :    for,  we  have 

0C=CB=BA-^AO. 

Hence,  the  sum  of  the  squares  of  the  diagonals  is  equiva- 
lent to  the  sum  of  the  squares  of  the  sides  (b.  iv.,  p.  1-i,  c.  2): 

that  is,  J^'+  OB^ o 4:AB^^ o 4 OB^' ; 

and  by  taking  away   OB^,  we  have, 

i~a'o3aiJ';   hence, 
AC^^    :     OB^     :     S     :     1;    or, 
AC    :     OB    ::     V^     :     1:  . 
hence,    the  side  of  the   inscribed  equilateral   triangle   is    to    the 
radius,  as  the  square  root  of  three,  to  unity. 


PROPOSITION  VI.     PROBLEM. 
In  a  given  circle   to  inscribe  a  regular  decagon. 

Let  0  be  the  centre,  and  OA   the    radius  of  the   given 
circle. 

Divide  the  radius  OA  in 
extreme  and  mean  ratio  at 
the  point  M  (b.  iv.,  pbob.  4) : 
Take  OM^  the  greater  seg- 
ment, and  lay  it  off  from  A 
to  B ;  the  chord  AB  will 
be  the  side  of  the  regu- 
lar decagon,  and  by  apply- 
ing it  ten  times  to  the  cir- 
cumference, the  decagon  will 
be  inscribed  in  the  circle. 

For,  drawing  MB,    we  have  by  construction, 
AO    :     OM    ::     OM    :     AM', 
or,  since  AB=OM, 

AO    I    AB    11    AB    I    AM, 


140 


GEOMETEY. 


But  since  the  triangles  ABO, 
AMB^  have  a  common  angle 
JL,  included  between  propor- 
tional sides,  they  are  similar 
(b.  IV.,  p.  20).  Xow  tlie  triangle 
BAO  being  isosceles,  AMB 
must  be  isosceles  also,  and 
AB  =  BM  ;  but  AB  =  OM ; 
hence,  also  MB=MO ;  hence, 
the  triangle  BMO  is  isosceles. 

Again,  in  the  isosceles  tri- 
angle BMO,  the  angle  AMB 
being  exterior,  is  double  the  interior  angle  0  (b.  i.,  p. 
25,  c.  6):  but  the  angle  AMB=MAB  \  hence,  the  triangle 
OAB  is  such,  that  each  of  the  angles  OAB  or  DBA,  at  its 
base,  is  double  the  angle  0,  at  its  vertex ;  hence,  the  three 
angles  of  the  triangle  are  together  equal  to  five  times  the  an- 
gle 0,  which  consequently,  is  the  fifth  part  of  two  right  angles, 
or  the  tenth  part  of  four;  hence,  the  arc  AB  is  the  tenth 
part  of  the  circumference,  and  the  chord  AB  is  the  side 
of  the  res'ular  decao"on. 

Cor.  1.  By  joining  the  vertices  of  the  alternate  angles 
of  the  decagon,  a  regular  pentagon  ACEGI  will  be  in- 
scribed. 

Cor.  2.  Any  regular  polygon  being  inscribed,  if  the  arcs 
which  the  sides  subtend  be  severally  bisected,  the  chords  of 
those  semi-arcs  will  form  a  new  regular  polygon  of  double 
the  number  of  sides  :  thus  it  is  plain,  that  the  square  will 
enable  us  to  inscribe,  successiveh^,  regular  polygons  of  8, 
16,  32,  ko,.,  sides.  And  in  like  manner,  by  means  of  the 
hexagon,  regular  polygons  of  12,  24,  48,  <Scc.,  sides  may  be 
inscribed ;  and  by  means  of  the  decagon,  polygons  of  20, 
40,  80,  <S:c.,  sides. 

Cor.  3.  It  is  further  evident,  that  any  of  the  inscribed 
polygons  will  be  less  than  the  inscribed  polygon  of  double 
the  number  of  sides,  since  a  part  is  less  than  the  whole. 


BOOK    V. 


141 


PKOPOSITION  VII.     PKOBLEM. 


A  regular  inscribed  iiohjrjon  heing  given,  to  circumscribe  a  sirri- 
liar  i^'^^ggon  about  the  same  circle. 

Let  0  be   the   centre   of    the   circle,    and    CDEFAB 
regular  inscribed  polygon. 

At  T,  the  middle 
point  of  the  arc  AB^ 
draw  a  tangent  GH^ 
and  do  the  same  at  the 
middle  point  of  each  of 
the  arcs  BC^  CD^  &c. ; 
these  tangents  will  be 
parallel  to  the  chords 
AB.BC,  CD,  &c.  (B.iiL, 
p.  10,  c.) ;  and  will,  by 
their  intersections,  form 
the  regular  circumscrib- 
ed polygon  GHIK  &c.,  similar  to  the  one  inscribed. 

For,  since  T  is  the  middle  point  of  the  arc  B2A,  and 
N  the  middle  point  of  the  equal  arc  BNC,  it  follows,  that 
BT=BN']  or  that  the  vertex  B  of  the  inscribed  polygon, 
is  at  the  middle  point  of  the  arc  A^BT.  Draw  OIL  The 
line   OH  will  pass  through  the  point  B. 

For,  the  right-angled  triangles  OTII,  NOII,  having  the 
common  hjqoothenuse  0/7,  and  the  side  0T=  ON,  are  equal 
(b.  I.,  P.  17),  and  consequently  the  angle  TOII=IIOX,  where- 
fore the  line  Oil  passes  through  the  middle  point  B  of  the 
arc  TN  (B.  ITT.,  p.  15).  In  the  same  manner  it  may  be 
shown  that  01  passes  through  (7;  and  similarly  for  the 
other  vertices. 

But    since  Gil  is   parallel   to  AB,  and  III  to  BC,    th 
angle  GUI  =  ABC  (b.  l,  p.  2-1);  in  like  manner,  IIIK=BCD 
and  so  for  the  other   angles :    hence,  the  angles  of  the  cir 
cumscribed    polygon    are   equal   to   those   of    the   inscribeeL 
And   further,  by  reason  of  these  same   parallels,  we  have 

Gil  :  AB  ::   Oil  :   OB,  and  III  :  BC  : :   OH  :   OB: 
therefore,  Gil    :     AB    :  :     HI    :    BC. 


142 


GEOMETRY. 


But  AB=BC, 
therefore  GE=1IL 
For  a  like  reason, 
UI=  IK^  ko,. ;  hence, 
the  sides  of  the  circum- 
scribed polj^gon  are  all 
equal;  hence,  this  poly- 
gon is  regular  and  simi- 
lar to  the  inscribed 
polygon. 

Cor.  1.  Eeciprocal- 
ly :  if  the  circumscribed  polygon  GHIK  &c.,  be  given,  and 
the  inscribed  one  ABC  &;c.,  be  required,  it  will  only  be 
necessary  to  draw  from  the  vertices  of  the  angles  6^,  H^  /, 
<fcc.,  of  the  given  polygon,  straight  lines  OG^  OH^  &c.,  meet- 
ing the  circumference  in  the  j^oints  A,  B.  (7,  ko,. ;  then  to 
join  these  points  by  the  chords  AB^  BC,  kc.  ;  this  Avill 
form  the.  inscribed  polygon.  An  easier  solution  of  this 
problem  Avould  be,  simph^  to  join  the  points  of  contact  T, 
A'J  P,  (fcc,  by  chords  2\VJ  iVP,  kc,  which  likewise  would 
form  an  inscribed  polygon  similar  to  the  circumscribed  one 

Cor.  2.  Hence,  we  may  circuniscribe  about  a  circle  any 
regular  polygon  similar  to  an  inscribed  one,  and  con- 
versely. 

Cor.  3.  It  is  plain  that  XR-hRT=RT+TG=ffG,  one 
of  the  equal  sides  of  the  polygon. 

Cor.  4.  If  through  B,  A,  F,  <fcc.,  the  middle  points  of 
the  arcs  XBT,  TAS,  SFB,  kc,  we  draw  tangent  lines,  we 
shall  thus  form  a  new  regular  circumscribed  polygon  having 
double  the  number  of  sides :  and  this  process  may  be  re- 
peated as  often  as  we  please.  The  new  polygon  will  be 
regular,  because  it  will  be  similar  to  a  new  inscribed  poly- 
gon which  may  be  formed  (p.  6,  c.  2)  of  double  the  number 
of  sides  of  the  first.  It  is  plain,  that  each  new  circumscribed 
polygon  will  be  less  than  the  one  from  which  it  was  den  ved, 
since  a  part  is  less  than  the  whole. 


BOOK   V. 


143 


FKOPOSITION  VIII.     THEOREM. 

The  area  of  a  regular  polygon  is  equal  to  its  perimeter  multi- 
plied hj  half  the  radius  of  the  inscribed  circle. 

Let  there  be  the  regular  polygon  GIIIK^  and  0^^^  OT, 
radii  of  the  inscribed  circle  drawn  to  the  .  points  of  tan- 
genc}^ :  then  will  its  area  be  equal  to  the  perimeter  mul- 
tiplied by  one-half  of  02\ 

For,  the  triangle  GOII  is 
measured  by  G1IX\0T\  the  trian- 
gle OHI,  by  HlXlOX:  but  0X= 
0T\  hence,  the  tw^o  triangles  taken 
together  are  measured  by 
{GH^-m)x\OT. 
And,  by  finding  the  measures  of 
the  other  triangles,  it  Avill  appear 
that  the  sura  of  them  all,  or  the 
whole  polygon,  is  measured  by  the  sum  of  the  bases  GH^ 
HI,  &c.,  or,  the  perimeter  of  the  polygon,  multiplied  by 
one-half  of  OT  \  that  is,  the  area  of  the  polygon  is  equal 
to  its  perimeter  multiplied  by  half  the  radius  of  the  in 
scribed  circle. 


PROPOSITION  IX.      THEOREM. 

The  perimeters  of  regular  polygons,  having  the  same  number  of 
sideSj  are  to  each  other  as  the  radii  of  the  circumscribed 
circles ;  and  also,  as  the  radii  of  the  inscribed  circles;  and 
their  areas  are  to  each  oilier  as  the  squares  of  those  radii. 

Let  AB  be  the  side  of  one  polygon,  0  the  centre,  and 
consequentl}^  OA  the  radius  of  the  circumscribed  circle, 
and  OB,  perpendicular  to  AB,  the  radius  of  the  inscribed 
circle.  Let  ah,  be  a  side  of  the  other 
polygon,  0  the  centre,  oa  and  od,  the 
radii  of  the  circumscribed  and  the 
inscribed  circles. 

Then,  the  perimeters  of  the  two 
polygons  are  to  each  other  as  the 
sides  AB  and  ah  (b.  IV.,  P.  27):  but 
the  angles  A  and  a  are  equal,  being 


lU 


GEOMETEY. 


each  half  of  the  angle  of  the  poly- 
gon ;  so  also  are  the  angles  i^  and 
b ;  hence,  the  triangles  ABO,  aho, 
are  similar,  as  are,  likewise,  the 
right.-angled  triangles  ADO,  ado; 
therefore. 


and  also,  as  the  radii 


AB  :  ab  ::  AO  :  ao  ::  DO  :  do  • 

hence,  the  perimeters  of  the  poly- 
gons are  to  each  other  as  the  radii 
AO,  ao,  of  the  circumscribed  circles 
DO,  do,  of  the  inscribed  circles. 

The  surfaces  of  these  polygons  are  to  each  other  as  the 
squares  of  the  homologous  sides  AB,  ab  (b.  lY.,  P.  27);  they 
are  therefore  likewise  to  each  other  as  the  squares  of  AO, 
ao,  the  radii  of  the  circumscribed  circles,  or  as  the  squares 
of  OD,  od,  the  radii  of  the  inscribed  circles. 


rEOPOSITIOX  X.     THEOEEM. 

Tico  regular  polygons,  of  the  same  number  of  sides,  can  always 
be  formed,  the  one  circumscribed  a.bout  a  circle,  the  other  in- 
scribed in  it,  which  shall  differ  from  each  oiJicr  by  less  tJian 
any  given  surface. 

Let  Q  be  the  side  of  a  square  less  than  the  given  sur- 
face. Bisect  AC,  a  fourth  part  of  the  circumference,  and 
then  bisect  the  half  of  this  fourth,  and  proceed  in  this 
manner,  always  bisecting  one  of  the  arcs  formed  by  the 
last  bisection,  until  an  arc  is  found  whose  chord  AB  is  less 
than  Q.  As  this  arc  will  be  an  exact  part  of  the  circum- 
ference, if  we  apply  the  chords  AB,  BC,  CD,  kc,  each  equal 
to  AB,  the  last  will  terminate  at  ^i,  and  there  will  be  formed 
a  regular  polygon  ABCDE  kQ.,  inscribed  in  the  circle. 

Next,  describe  about  the  circle  a  similar  polygon  abcds 
&c.  (p.  7)  :  the  difference  of  these  two  polygons  will  be 
less  than  the  square  of  Q. 

For,  from  the  points  a  and  b,  draw  the  lines  aO,  bO,  to 
the  centre  0:  they  will  pass  through  the  points  A  and  B 
(p.  7).     Draw  also  OK  to  the  point  of  contact  K:   it  will 


BOOK    V, 


145 


bii^ect  AB  in  I,  and  be  per- 
pendicular to  it  (b.  hi.,  p.  6,  s.) 
Prolong  AO  io  E^  and  draw 
BE. 

Let  p  represent  the  cir- 
cumscribed polygon,  and  P  the 
inscribed  polygon:  then  since 
the  triangles  aOh^  A  OB,  are 
like  parts  of  p  and  P,  we 
have  (b.  ii.,  p.  11), 

aOb    :     AOB    ::     p     :    P: 

But  the  triangles  being  similar  (b.  iv.,  P.  25), 


a  Oh 


Hence, 


AOB    ; 
p     :    P 


Oa 
0~a 


0A\  or  OK 
6K\ 


Again,  since  the  triangles  OaK^  EAB  are  similar,  having 
their  sides  respectively  parallel  (b  iv.,  p.  21). 

Oil      :     OZ'    :  :     AE"     :     EB',  hence 

p     :    P    :  :     AE"    :     EB"^    or  by  division  (b.  ii.,  p.  6), 

p    :     p-P  ::     Ilf    :     AE"" ~ EB' ,  or  AB''\ 

But  p  is  less  than  the  square  described  on  the  diameter 
^^(p.  7,  C.4);  therefore,  p  —  P\s  less  than  the  square  de- 
scribed on  AB :  that  is,  less  than  the  given  square  on  Q: 
hence,  the  difference  between  the  circumscribed  and  inscribed 
poh^gons  may,  by  increasing  the  number  of  sides,  always  be 
made  less  than  any  given  surface. 


PEOPOSITION    XI.     PROBLEM. 

The  surface  of  a  regular  inscrihed  polygon^  and  that  of  a  sim- 
ilar circumscribed  polygon^  being  given  ;  to  find  the  surfaces 
of  the  regular  inscrihed  and  circumscrihed  polygons  having 
double  the  number  of  sides. 

Let  AB  be  a  side  of  the  given  inscribed  polygon ; 
EF,  parallel  to  AB,  a  side  of  the  circumscribed  polygon, 
and  C  the  centre  of  the  circle.  If  the  chord  AM  and  the 
tangents  APj  BQ,  be  drawn,  AM  will  be  a  side  of  an  in- 

10 


146 


GEOMETRY. 


scribed  polj^gon,  having  twice  the  number  of  sides;  and 
AP  +  PM=2PM  or  Pft  will  be  a  side  of  the  similar  cir- 
cumscribed  polygon  (p.  7,  c.  3). 

Now,  as  the  same  construc- 
tion will  take  place  at  each 
angle  corresponding  to  ACM^ 
It  will  be  sufficient  to  consider 
ACM  by  itself;  for  the  trian- 
gles connected  with  it  are  evi- 
dently to  each  other  as  the 
whole  polygons  of  which  they 
form  part.  Let  P,  then,  be 
the  surface  of  the  inscribed 
polygon  whose  side  is  AB^  p,  that  of  the  similar  circum- 
scribed polygon;  P'  the  surface  of  the  polygon  whose  side 
is  AM^  p  that  of  the  similar  circumscribed  polygon :  P 
and  p  are  given ;    we  have  to  find  P  and  p'. 

First.  Now  the  triangles  ACP,  ACM,  having  the  com- 
mon vertex  J.,  are  to  each  other  as  their  bases  CD,  CM  (b. 
IV.,  P.  6,  c.) ;  they  are  likewise  to  each  other  as  the  poly- 
gons P  and  P',  of  which  they  form  part  (b.  II.,  P.  11) :  hencCj 

P    :     P'  ::     CD     :     CM. 
Again,  the  triangles  CAM,  CME,  having  the  common  vertex 
M,  are  to  each  other  as  their  bases  CA,  CE ;    they  are  like- 
wise to  each  other  as  the  polj'gons  P'  and  p  of  which  they 
form  part ;    hence, 

P'     :    p     ::     CA     :     CR 
But  since  AD  and  MF  are  parallel,  we  have, 

CD     :     CM    :  :     CA     :     CF- 
h«nce,  P    :     P'     ::     P'     :    p; 

hence,  the  polygon  P'  is  a  mean  proportional  between  the 
two  given  polygons  P  and  p,  and  consequently, 

P'  =  VP^. 
Secondly.   The  altitude  CM  being   common,    the  triangle 
CPM  is   to  the  triangle  CPF,  as  PJ/ is  to  PE',   but  since 
CP  bisects   the  angle  MCE,  we  have  (b.  iv.,  p.  17), 
PM  :    PE   ::     CM   :     CE    :  :     CD    :     CA    :  :    P    :    P'; 
hence,  CPM    :     CPE    :  :     P    :     P' ; 


BOOK    y.  147 

and  consequently, 

CPM    :     CP2I+CPE,  OT  CUE    ::     P    :     P+P'-, 
and  hence,  2CPM,  or  CMPA    :     CME    :  :     2P    :    P+P*. 

But  CMPA  is  to  6'JZE'  as  the  polj^gons  p'  and  79,  of  ^vhich 
hey  form  part:    hence, 

p'     :    p     ::     2P    :     P-\-P\ 

Now  as  P'  has  been  already  determined ;  this  new  propor- 
tion will  serve  to  determine  p\  and  give  us 

and  thus  by  means  of  the  polygons  P  and  p  it  is  easy  to 
find  the  polygons  P'  and  p'^  which  have  double  the  num- 
ber of  sides. 


PEOrOSITION   XII.     PEOBLEM. 
To  find  the  approximate  area  of  a  circle  ichose  radius  is  unity. 

Let  the  radius  of  the  circle  be  1 ;  the  side  of  the  in- 
scribed square  will  be  ^/2  (p.  3,  S.) ;  that  of  the  circum- 
scribed square  will  be  equal  to  the  diameter  2 ;  hence,  the 
surface  of  the  inscribed  square  will  be  two,  and  that  of  the 
circumscribed  square  will  be  4,  Ilence,  P=2,  and  ^J=4 ; 
by  the  last  proposition  we  shall  find  the 

inscribed  octagon     P'=  v/8  =  2.828427l, 

circumscribed  octagon    p'=-—- — 73=3.3137085. 

^+  V  o 

The  inscribed  and  the  circumscribed  octagons  beim?  thus 
determined,  we  shall  easily,  by  means  of  them,  determine 
the  polygons  having  twice  the  number  of  sides.  We  have 
only  in  this  case  to  put  P=2.828427l,  2?=3.3137085  ;  we 
shall  find 

P'=  '/Pxi>-3.0614674, 

2PX  77 

i/=p^-;=3.1825979. 

These  polygons  of  16  sides  will  enable  us  to  find  the 
polygons  of  32  sides ;  and  the  processes  may  be  continued 


us 


GEOMETRY. 


until  llie  diflercijce  between  tlie  inscribed  and  circumscribed 
polygons  is  less  than  any  given  surface  (p.  10).  Since  the 
circle  lies  between  the  polygons,  it  will  differ  from  either 
polygon  by  less  than  the  polygons  differ  from  each  other: 
and  hence,  in  so  far  as  the  figures  which  express  the  areas 
of  the  two  polygons  agree,  they  will  be  the  true  figures  to 
express  the  area  of  the  circle. 

We  have  subjoined  the  computation  of  these  polygons, 
carried  on  till  they  agree  as  far  as  the  seventh  place  of 
decimals. 


Number  of  Sides. 

Inscribed  Polygons.                     Circumscribed  Polygons 

4       . 

2.0000000      .      .      . 

4.0000000 

8       . 

2.8281271      .      .      . 

8.3137085 

16       .      . 

8.0614674      .      .      . 

8.1825979 

82       .      . 

8.1214451      .       .       . 

8.1517249 

64      .      . 

3.1365485      .       .      , 

8.1441184 

128       .      . 

8.1403311      .       . 

8.1422236 

256       .      , 

3.1412772      .      . 

8.1417504 

512       . 

8.1415138      .       . 

8.1416321 

102-1       . 

3.1415729      .       . 

8.1416025 

2048       .      , 

8.1415877      .      .      . 

8.1415951 

4096       . 

.      .        8.1415914      .      .      . 

8.1415938 

8192       . 

.        3.1415923      .       .      . 

8.1415928 

16384       . 

.      .       3.1415925      .      . 

8.1415927 

82768       . 

.       .       3.1415926      .      . 

8.1415926 

The  approxi 

mate  area  of  the  circle,  we 

infer,  therefore, 

is  equal  to  8.1415926.  Some  doubt  may  exist  perhaps 
about  the  last  decimal  figure,  owing  to  errors  proceeding 
from  the  parts  omitted  ;  but  the  calculation  has  been  car- 
ried on  with  an  additional  figure,  that  the  final  result  here 
given  might  be  absolutely  correct  even  to  the  last  decimal 
])lace.  The  number  generally  used,  for  computation,  is 
8.1416,  a  number  very  near  the  true  area. 

Scholium  1.  Since  the  inscribed  polygon  has  the  same 
number  of  sides  as  the  circumscribed  polygon,  and  since 
the  two  polygons  are  regular,  they  will  be  similar  (p.  1) : 
and,  therefore,  when  their  areas  approach  to  an  equality 
with  the  circle,  their  perimeters  will  approach  to  an  equal- 
ity with  the  circumference. 


BOOK  y. 


149 


Scholium  2.  That  magnitude  to  wliicli  a  varying  mag- 
nitude approaches  continually,  and  which  it  cannot  pass,  is 
called  a  limit. 

Having  shown  that  the  inscribed  and  circumscribed 
polygons  may  be  made  to  differ  from  each  other  by  less 
than  any  given  surface  (p.  10),  and  since  each  differs  from 
the  circle  less  than  from  the  other  polygon,  it  follows  that 
the  circle  is  the  limit  of  all  inscribed  and  circumscribed 
polygons,  formed  by  continually  doubling  the  number  of 
sides,  and  that  the  circumference  is  the  limit  of  their  peri- 
meters. Hence,  no  sensible  error  can  arise  in  supposing 
that  what  is  true  of  such  a  polygon  is  also  true  of  its 
limit,  the  circle.  Indeed,  the  circle  is  but  a  regular  poly- 
gon of  an  infinite  number  of  sides. 

TKOPOSITION   XIII.     THEOKEM. 


The    circumferences    of  circles    are   to  each  other  as  tlieir  radii^ 
and  the  areas  are  to  each  other  as  tlie  squares  of  their  radii. 

Let  us  designate  the  circumference  of  the  circle  whose 
radius  is  CA  by  circ.  CA  ;  and  its  area,  by  area  CA  :  it  is 
then  to  be  shown  that 


circ.   CA 
area  CA 


circ.   OB 
area  OB 


CA 
CA' 


OB,  and  that 

olr. 


Inscribe  within  the  circles  two  regular  polygons  of  the 
same  number  of  sides.  Then,  whatever  be  the  number  of 
sides,  their  perimeters  will  be  to  each  other  as  the  radii 
CA  and  Oi>  (p.  9).     Now,  i:'  the  arcs  subtended  by  the  sides 


150 


GEOMETKY 


of  the  polygons  be  continually  bisected,  and  corresponding 
polygons  formed,  the  perimeter  of  each  new  polygon  will 
approach  the  circumference  of  the  circumscribed  circle,  and 
at  the  limit  (p.  12,  s.  2),  we  shall  have 

circ.   CA     :     cur.   OB    :  :     CA     :     OB. 


Again,  the  areas  of  the  inscribed  polygons  are  to  each 
other  as  CA  to  OB'  (p.  9).  But  when  the  number  of 
sides  of  the  polygons  is  increased,  as  before,  at  the  limit 
we  shall  have 


area  CA 


area  OB 


CA' 


OB' 


Cor.  1.  It  is  plain  that  the  limit  of  any  portion  of  the 
perimeter  of  an  inscribed  regular  polygon  lying  between 
the  vertices  of  two  angles,  is  the  corresponding  arc  of  the 
circumscribed  circle.  Thus,  the  limit  of  the  portion  of  the 
perimeter  intercepted  between  G  and  ^  is  the  arc  GJ^'B. 

Cor.  2.  If  we  multiply  the  antecedent  and  consequent 
of  the  second  couplet  of  the  first  proportion  by  2,  and  of 
the  second  by  4,  we  shall  have 

circ.   CA     :     circ.    OB     ::     2CA     :     2 OB; 
and  area  CA     :     area  OB     ::     46ll'    :     46^; 

that  is,  the  circumferences  of  circles  are  to  each  other  as  iherr 
diameters^  and  their  areas  are  to  each  other  as  the  squares  Oj 
iJ^Leir  diameters. 


BOOK    Y. 


151 


PKOPOSITION   XIV.      TIIEOKEM. 

Simihr  arcs  are  to  each  other  as  their  radii:   and  similar  sec- 
tors are  to  each  oilier  as  the  squares  of  their  radii. 

Let  AB,  BE,  be   similar  arcs,  and  ACB^  BOB,  sinuJar 
sectors :    then 

•    AB    ',     BE    \\ 
and  ACB    :     BOB 


CA 


OB 


CA' 


0B\ 


For,  since  the  arcs  are  sim- 
ihxr,  the  angl^  C  is  equal  to 
the  angle  0  (b.  I  v.,  d.  (3).  But 
we  have   (b.  hi.,  p.  17), 

angle  G  :  4  right  angles 
and,  angle  0  :  4z  right  angles 
hence  (b.  il,  p.  4,  c), 

AB    :     BE    ::     circ. 


CA     :     circ.   OB] 


but  these   circumferences   are  as  the  radii  AC^  BO  (p.  13); 
hence, 

AB    :     BE    ::     CA     :     OB, 

For  a  like  reason,  the  sectors  ACB^  BOE,  are  to  each 
other  as  the  Avhole  circles :  which  again  are  as  the  squares 
of  their  radii  (p,  13) ;    therefore, 


sect.  ACB    :     sect.  BOB 


CA' 


OB". 


PROPOSITION   XV.      THEOREM. 


TJie  area  of  a  circle  is  equal  to  tlte  product  of  half  the  radius 
by  tlte  circumference. 


Let  ACBE  be  a  circle  whose  cen- 
^0   is   0  and  radius   OA  :    then  will 


area  OA=lOAXcirc. 


OA. 


For,  inscribe  in  the  circle  any  regu- 
lar polygon,  and  draw  OE  perpendicu- 
lar to  one  of  its  sides.      The  area  of 


152 


GEOMETRY. 


the  polygon  is  equal  to  ^OF,  mul- 
tiplied by  the  perimeter  (p.  8).  Now, 
let  the  arcs  which  are  subtended  by 
the  sides  of  the  polygon  be  bisected 
and  new  polygons  formed  as  before ; 
the  limit  of  the  perimeter  is  the  cir- 
cumference of  the  circle  ;  the  limit  of 
the  apothem  is  the  radius  OA,  and 
the  limit  of  the  area  of  the  polygon  is  the  area  of  the 
circle  (p.  12,  s.  2).  Passing  to  the  limit,  •  the  expression  for 
the  area  becomes 

area   OA=^OAXcirc.   OA  ; 
consequently,  the  area  of  a  circle   is   equal  to  the  product 
of  half  the  radius   by   the   circumference. 

Cor.    The   area  of  a   sector   is   equal   to   the   arc  of  the 
sector  multiplied  by  half  the  radius. 
For,  we  have  (b.  hi.,  p.  17,  s.  4), 

sect.  A  CB :  area  CA  :  :  AMB  :  circ.  CA  ; 

or,  sect.  A  CB    :    area  CA   : :    AMB  X 

iCA   :   circ.  CAxiCA. 

But,  circ.  CAx^CA  is  equal  to  the 
area  CA  ;  hence,  AMBx^CA  is  equal 
to  the  area  of  the  sector. 


PPvOPOSITIOX  XVI.      TIIEOKEM. 

The  area  of  a  circle  is  equal  to  the  square  of  the  radius  muh 
tiplied  hy  the  ratio  of  the  diameter  to  the  circumference. 

Let  the  circumference  of  the  circle  whose  diameter  is 
unity  be  denoted  by  -tt  :  then,  since  the  diameters  of  cir- 
cles are  to  each  other  as  their  circumferences  (p.  13,  c.  2),  if 
will  denote  the  ratio  of  any  diameter  to  its  circumference. 
We  shall  then  have 

Ire::     2CA     :     circ.   CA  : 
therefore,  circ.   CA^^rrXlCA. 

Multiplying  both  members  by  \CA.  we  have 
\CAXcirc.CA-=^XCX\ 


BOOK    V 


loi> 


or  (p.  15)         area  CA=^xCA\ 

tliat   is,    tJte   area    of  a    circle  is  equal 

to  '^  into  the  square  of  the  radius. 

Scholium  1.  Let  OA  =  R,  and  area 
CA  —  A  :     then,     A  =  '^W-^     making 
CA  =  1]  we  shall  have 
area  CA='n'. 

But  we  have  found  the   area  of  the  cirele  whose  ladius  is 
1  to  be  8.1415926  (p.  12):    therefore,  we  have 

^=3.1415926. 

In  common  calculations,  we  take   -^=3.1-416. 

Scholium  2.  The  problem  of  the  quadrature  of  the  circle, 
as  it  is  called,  consists  in  finding  a  square  equivalent  in 
surface  to  a  circle,  the  radius  of  which  is  known.  Now  it 
has  just  been  proved,  that  a  circle  is  equivalent  to  the  rect- 
angle contained  by  its  circumference  and  half  its  radius 
(p.  15);  and  this  rectangle  maj^  be  changed  into  an  equiv- 
alent square,  by  finding  a  mean  proportional  between  its 
length  and  its  breadth  (b.  iv.,  pkob.  3).  To  square  the 
circle,  therefore,  is  to  find  the  circumference  when  the 
radius  is  given ;  and  f  )r  efiecting  this,  it  is  enough  to 
knoAV  the  ratio  of  the  diameter  to  the  circumference. 

Ilitlierto  the  ratio  in  question  has  never  been  determin- 
ed except  approximatively  ;  but  the  aj^proximation  has 
been  carried  so  far,  that  a  knowledge  of  the  exact  ratio 
would  afford  no  real  advantage  whatever  beyond  that  of 
the  approximate  ratio.  Accordingly,  this  probUnn,  which 
engaged  geometers  so  dee])ly,  when  their  methods  of 
approximation  were  less  perfect,  is  now  degraded  to  the 
rank  of  those  idle  questions,  with  which  no  one  possessing 
the  slightest  tincture  of  geometrical  science,  will  occupy 
any  portion  of  his  time. 

Archimedes  showed  that  the  ratio  of  the  diameter  to  the 
circumferenos  is  included  between  3fJ  and  35?  ;  hence,  3} 
or  2f  affords  at  once  a  pretty  accurate  approximation  to 
the  number  above  designated  by  -r ;  and  the  simplicity 
of  this  first  approximation  has  brought  it  into  veiy  general 


154 


GEOMETKY. 


use.  Mdius^  for  the  same  quantity,  found  the  much  more 
accurate  value  fff.  At  last,  the  value  of  -r,  developed  to 
a  certain  order  of  decimals,  was  found  by  other  calculators 
to  be  3.1-il592G535897932  kc.\  and  some  have  had  patience 
enough  to  continue  these  decimals  to  the  hundred  and 
twenty -seventh,  or  even  to  the  hundred  and  fortieth  place. 
Such  an  approximation  is  practically  equivalent  to  perfect 
accuracy :  the  root  of  an  imperfect  power  is  in  no  case 
more  accurately  known. 

PE0P09ITI0X  XVII.     TIIEOKEM. 

If  the  circumferences  of  two  circles  intersect  each  other,  the  arc 
of  the  common  chord  in  the  less  circle  will  he  lorujer  than 
tlte  corres])onding  arc  of  the  greater!^ 

Let  A  and  B  be  the  centres  of  two  circles,  AC^  BC^ 
their  radii,  C  and  D  the  points  in  which  their  circumfer- 
ences intersect  and  CD  their  common  chord :  then  will  the 
arc  DEC  described  with  the  radius  BC^  be  longer  than  the 
arc  DFC  described  with  the  greater  radius  AC, 

Join  the  centres  A  and  i?, 
and  prolong  AB  to  E.  Then, 
since  AB  bisects  the  chord 
CD  at  right  angles  (b.  iil,  p.  11) ; 
it  also  bisects  the  arcs  at 
the  points  F  and  E  (b.  hi.,  p. 
6).  Draw  CE  and  DE  which 
will  be  equal  to  each  other 
(b.  III.,  P.  4) ;    also  CF  and  DF. 

Bisect  the  arcs  CE^  ED^ 
and  also  the  arcs  CF^  FD^  and 
draw  chords  subtending  the  new 
arcs :  there  will  thus  be  inscribed  in  the  two  segments 
DEC^  DFC^  portions  of  two  polygons,  having  the  same 
number  of  sides  in  each. 

Now,  since  the    point   F   is    within   the   triangle   DEO^ 


*  The   arc   considered  in   this   demonstration  is  the   one  wliich  is  less  thiiu 
semicircle. 


BOOK    Y.  155 

EC  plus  ED  is  greater  tlian  CF  plus  ED  (b.  [.,  p.  8) : 
hence,  the  half,  CE  is  greater  than  the  half,  CE.  If  now, 
with  (7  as  a  centre,  and  CE  as  a  radius,  we  describe  an 
arc  EIT^  the  chord  CE  being  greater  than  CE,  the  arc  CEH 
will  be  gi-eater  than  the  arc  CE  (b.  hi.,  p.  5).  If  we  sup- 
pose the  arc  CKE  to  move  with  the  chord  CE  then, 
when  the  chord  CE  becomes  the  chord  Clf^  the  arc  CKE 
will  pass  through  the  points  C  and  11^  and  will  have  with 
CEII^  the  common  chord   CII. 

If,  now,  we  bisect  the  arc  which  is  equal  to  CKE,  and 
also  the  arc  CFII,  we  know  from  what  has  already  been 
shown,  that  the  chord  of  half  the  outer  arc  will  be  greater 
than  the  chord  of  half  the  inner  arc  CEII,  much  more  will 
it  be  greater  than  the  chord  of  6X,  which  is  half  the  arc 
CE '^  that  is,  the  chord  of  the  arc  CK^  one-half  of  CE^ 
will  always  be  greater  than  the  chord  of  the  arc  CL^  one- 
half  of  CE.  Hence,  the  perimeter  of  that  portion  of  the 
polygon  inscribed  in  the  segment  CED,  will  be  greater  than 
the  perimeter  of  the  corresponding  polygon  inscribed  in  the 
segment  CED.  If,  then,  we  continue  the  operations  indefi- 
nitely, the  limit  of  the  outer  perimeter  will  be  the  arc  CED, 
and  of  the  inner,  the  arc  CED:  hence,  the  arc  CED  is 
greater  than  the  arc   CED. 

Cor.  If  equal  chords  be  taken  in  unequal  circles,  the 
arc  of  the  chord  in  the  greatest  circle  will  be  the  shortest ; 
for,  the  circles  may  always  be  placed  as  in  the  figure. 


BOOK    VI. 


PLANES    AND    TOLYEDRAL  ANGLES.. 


DEFINITIONS. 


1.  A  straight  line  is  'perj)endicular  to  a  iiilane,  when  it 
is  perpendicular  to  every  straight  line  of  the  ph  le 
which  passes  through  its  foot  :  converseh^,  the  plane  is 
perpendicular  to  the  line.  The  point  at  which  the  perp  n- 
dicular  meets  the  plane,  is  called  \\iq  foot  of  the  perpencic- 
ular. 

2.  A  line  is  'parallel  to  a  pkme,  when  it  cannot  meet 
that  plane,  to  what  distance  soever  both  be  produced. 
Conversely,  the  plane  is  parallel  to  the  line. 

3.  Two  planes  are  parallel  to  each  other,  when  they 
cannot  meet,  to  wdiat  distance  soever  both  be  produced. 

4.  The  indefinite  space  included  between  tw^o  planes 
which  intersect  each  other,  is  called  a  di^dral  angle:  the 
planes  are  called  the  faces  of  the  angle,  and  their  line  of 
common  intersection,  the  edge  of  the  angle. 

A  diedral  angle  is  measured  by  the  angle  contained  be- 
tween two  lines,  one  drawn  in  each  fiice,  and  both  perpen- 
dicular to  the  common  intersection  at  the  same  point.  This 
angle  may  be  acute,  obtuse,  or  a  right  angle.  If  it  is  a 
right  angle,  the  two  faces  are  perpendicular  to  each  other. 

5.  A  PoLYEDRAL  angle  is  the  indefinite  space  incjluded 
by  several  planes  meeting  at  a  common  point.  Each  plane 
is  called  a  face:  the  line  in  which  any  two  faces  intersect, 
is  called  an  edge:  and  the  common  point  of  meeting  of  all 
the  planes,  is  called  the  vertex  of  the  polled ral  angle. 


BOOK    YI.  •  157 

Th'"  ?,  the  poljedral  angle  wliose  ver-  g 

lex  is   %  is   bounded   by  the   four  faces,  y/j\ 

ASD,  jISO,  CSD,  DSA.     Three  planes,  at  /7     \\ 

least,  are  necessary  to   form  a  polyedral  ^//- /-Jc 

angle.  //        1/ 

A  polyedral  angle  bounded  by  three  Z^ / 

planes,  is  called  a  tiiedral  angle. 

POSTULATES. 

1.  Let  it  be  granted,  that  from  a  given  point  of  a 
plane,  a  line  may  be  drawn  perpendicular  to  that  plane. 

2.  Let  it  be  granted,  that  from  a  given  point  without  ji 
plane,  a  perpendicular  may  be  let  fall  on  the  plane. 

rROPOSITION  I.     THEOREM. 

A  straight  line  cannot  he  imrthj  in  a  plane^  and  partly  out  of  it. 

For,  by  the  defmition  of  a  plane  (b.  I.,  D.  9),  Avhen  a 
straight  line  has  two  points  common  with  it,  the  line  lies 
wholly  in  the  plane: 

Scholiam.  To  discover  Avhether  a  surface  is  plane,  apply 
a  straight  line  in  dift'erent  ways  to  that  surface,  and  ascer- 
tain if  it  coincides  Avith  the  surface  throughout  its  whole 
extent. 

rROPOSITION  II.     THEOREM. 

Two  straight   lines    ivhich   intersect  each  other^  lie  in    the   same 
2)lane,  and  dderniine  its  position. 

Let  AB^  AC^  be  two  straight  lines 
which  intersect  each  other  in  ^  ;  a  plane 
may  be  conceived  in  which  the  straight 
line  AB  is  found  ;  if  this  plane  be  turned 
round  AB,  until  it  pass  through  the  point 
(7,  then  the  line  AC^  which  has  two  of  its  points  A  and 
(7,  in  this  plane,  lies  wholl}^  in  it ;  hence,  the  position  of 
the  plane  is  determined  by  the  single  condition  of  contain- 
ing the  two  straight  lines  AB,  AC. 


158 


GEOMETRY, 


Cor.  1.  Any  three  points  ^,  i?, 
C,  not  in  a  straight  line,  determine 
the  position  of  a  pkne.  Ilence,  a 
triangle  BAC,  determines  the  posi- 
tion of  a  plane. 

6>'r.  2.  Ilence,  also,  two  paral- 
lel? AB.,  CD^  determine  the  posi- 
tion of  a  plane  ;  for,  drawing  the 
secant  EF^  the  plane  of  the  two 
straight  lines  AE^  EF^  is  that  of 
the  parallels  AB,  CD.  But  the  lines  AE,  EF,  determine 
this  plane ;    therefore,  so  do  the  parallels,  AB^   CD. 


C 


/F 


D 


rKOPOSITIOX  III.     THEOREM. 

If  two  ^;?a?ze.s  cut  one  another^  their   common  section  loill  be  a 
straiglit  line. 

Let  the  two  planes  AB^  CD^  cut 
one  another,  and  let  E  and  F 
be  two  points  of  their  common 
section.  Draw  the  straight  line 
EF.  This  line  lies  wholly  in  the 
plane  AB^  and  also,  wholly  in  the 
plane  CD  (b.  i.,  d.  9) :  therefore, 
it  is  in  both  planes  at  once.  But 
since  a  straight  line  and  a  point  out  of  it  cannot  lie  in 
two  planes  at  the  same  time  (p.  ii.,  c.  1),  EF  contains  all 
the  points  common  to  both  planes,  and  consequently,  is 
their  common  intersection. 

rKOPOSITIOX  IV.     THEOREM. 

Lf  a  straiglit  line  he  ijerpendicular  to  two  straight  lines  at  ihcir 
point  of  intersection^  it  will  he  j^^^J^^ndicular  to  the  plane 
of  those  lines. 

Let  J/iV  be  the  plane  of  the  two  lines  BL\  CC,  and  let 
AP  be  perpendicular  to  each  of  them  at  their  point  of 
intersection  P ;  then  will  AP  be  perpendicular  to  every 
line  of  the  plane  passing  through  P,  and  consequently  to 
the  plane  itself   (u.  1). 


BOOK    YL 


159 


For,  throiigli  P  draw  in  the 
plane  il//V,  any  straight  line  as  PQ. 
Through  any  point  of  this  line,  as 
Q,  draw  BQO,  so  that  BQ  shall 
be  equal  to  QC  (b.  IV.,  prob.  5); 
draw  AB,  AQ,  AC. 

The  base  BC  being  divided  into 
two  equal  parts  at  the  point  ft  the 
triangle  BPC  gives  (B.  iv.,  P.  14:). 

PC^+PB''=c>=2PQ'+2QO\ 
The  triangle  BAG  in  like  manner  gives, 

AC^'+AB^^  =0=210" -\-2QC\ 
Taking  the  first  of   these   equals  from  the  second,  and 
observing  that  the  triangles  APC,  APB,  being  right-angled 
at  P,  give 

J^'-P^'=o=AP'',  and  AB^' - PB' =o AP\ 
we  shall  have, 

AP^+AP"  o  2AC/-2PQ', 
Therefore,  by  taking  the  halves  of  both,  we  have 

AP^^^oAQ'-PQ'^  or  AQ'^<o=AP'-\-PQ^ ; 
hence,  the  triangle  APQ   is   right-angled  at  P;  hence,  AP 
is  perpendicular  to  PQ. 

Srholium.  Thus,  it  is  evident,  not  only  that  a  straight 
line  may  be  perpendicular  to  all  the  straight  lines  which 
pass  through  its  foot,  in  a  plane,  but  that  it  always  must 
be  so,  whenever  it  is  perpendicular  to  two  straight  lines 
drawn  in  the  plane :  hence,  a  line  and  plane  may  fulfil  the 
conditions  of  the  first  definition.  * 

Cor.  1.  The  perpendicular  AP  is  shorter  than  any 
oblique  line  y1  $  ;  therefore,  it  measures  the  shortest  distance 
from  the  point  A  to  the  plane  J/IV. 

Cor.  2.  At  a  given  point  P,  on  a  plane,  it  is  impossi- 
ble to  erect  more  than  one  perpendicular  to  the  plane ;  for, 
if  there  could  be  two  perpendiculars  at  the  same  point  P, 
draw  through  these  two  perpendiculars  a  plane,  whose  sec- 
tion   with  the  plane  MN  is  PQ\    then   these   tw-o    perpen- 


160 


GEOMETBY. 


diculars   Vs'oald   be   "botli   pcritendiciilar   to  the  line  PQ,  at 
the  same  point,  Avliicli  is  impossible  (b.  i.,  p.  14,  c.) 

It  is  also  impossible  to  let  fall  from  a  given  point,  out 
of  a  i)lanc,  two  perpendiculars  to  tliat  plane ;  for,  if  AP, 
AQ^  be  two  such  perpendiculars,  the  triangle  APQ  Avill 
have  two  right  angles  APQ^  ^QP)  which  is  impossible  (b. 
L,  P.  25,  c.  8). 


PROPOSITION  V.     THEOEEM. 

If^  from  a  point  witlioiit  a  j^^cme,  a  jierpendicular  he  drawn  to 
the  plane,  and  ohliqiLe  lines  he  draicn  to  its  differefiit points'. 

\st.    The  ohlique  lines  which    meet    the  plane  at  points  equally 
distant  from  the  foot  of  the  p)erpendicular^  are  equal : 

2d.    Of  two  ohlique  lines  icldch    meet    the  p)lane  at  unequal  di'i- 
tances^  the  one  passinrj  through  the  remote  point  is  tJie  longer. 

Let  AP  be  perpendicular  to  the  plane  MN\  AB,  ACy 
AD,  oblique  lines  intercepting  the  equal  distances  PB,  POj 
PD^  and  AE  a  line  intercepting  the  larger  distance  PE\ 
then  will  AB=AC=^AB\  and  AE  will  be  greater  than 
AB. 

For,  the  angles  APB,  APC,  APB, 
being  right  angles,  and  the  distances 
PB,  PC,  PB,  equal  to  each  other, 
the  triangles  APB,  APC,  APB, 
have  iin  each  an  equal  angle  con- 
tained by  two  equal  sides  :  there- 
fore they  are  equal  (b.  i.,  p,  5) ; 
hence,  the  hypothenuses,  or  the 
oblique  lines  AB,  AC,  AB,  are  equal 
to  each  other. 

Again,  since  the  distance  PE  is  greater  than  PB,  or  its 
equal  PB,  the  oblique  line  AE  is  greater  than  AB,  or  its 
equal  AB  (b.  1,  P.  15). 

Cor.  All  the  equal  oblique  lines,  AB,  AC,  AB,  kc,  ter- 
minate in  the  circumference  BCB,  described  from  P,  the 
foot  of  the  perpendicular,  as  a  centre ;  therefore,  a  point  A 
being  given  out  of  a  plane,  the  point  P  at  which  the  per- 


BOOK    YI.  161 

pendicular  let  fall  from  A  would  meet  tliat  plane,  maj  be 
found  by  marking  upon  that  plane  three  points,  B,  C,  Z>, 
equally  distant  from  the  point  J.,  and  then  finding  the 
centre  of  the  circle  which  passes  through  these  points ;  this 
centre  will  be  P,  the  point  sought. 

/Scholium.  The  angle  ABP  is  called  the  vicUnaiion  of  the 
obliqiie  line  AB  to  the  plane  J/xY;  which  inclination  is  evi- 
dently equal  with  respect  to  all  such  lines  AB^  AC,  AD,  as 
make  equal  angles  with  the  perpendicular ;  for,  all  the  tri- 
angles ABP,  ACP,  ABP,  &c.,  are  equal  to  each  other. 

PEOPOSITION    v^I.     TIIEOEEM. 

If  from  the  foot  of  a  perpendicular  a  line  he  drawn  at  right 
angles  to  any  line  of  a  plane,  and  the  point  of  intersection 
he  joined  ivith  any  point  of  the  perpendicular,  this  last  line 
luill  he  perpendicular  to  the  line  of  the  plane. 

Let  JIP  be  perpendicular  to  the  plane  NM,  and  PD 
perpendicular  to  BC ',  join  B  with  any  point  of  the  per-" 
pendicular,  as  A  ;  then  will  AB  also  be  perpendicular  to 
BG. 

TiJ^e  DB=BC,  and  draw  PB, 
PC,  AB,  AC.  Kow,  since  DB  is 
equal  to  DC,  the  oblique  line  PB  M^ 
is  equal  to  PC  (b.  1,  p.  5) :  and 
since  PB  is  equal  to  PC,  the 
oblique  line  AB  is  equal  to  AC 
(p.  5)  ;  therefore,  the  line  AD 
has    two    of    its    points   A   and  D  \  / 

equally   distant  from    the   extremi-  ^' 

ties  B  and  (7;  therefore^  AD  is  a  perpendicular  to  BC,  at 
its  middle  point  D  (b.  i.,  p.  16,  c.) 

Cor.  It  is  evident,  likewise,  that  BC  is  perpendicular 
to  the  plane  of  the  triangle  APD,  since  it  is  perpendicu- 
lar to  the  two   straight   lines  AD,  PD  of  that  plane  (p.  4). 

Scholium  1.  The  two  lines  AE,  BC,  afford  an  instance  of 
two  lines  which  are  not  parallel,  and  yet  do  not  meet,  be- 
cause they  are  not  situated  in  the  same  plane.     The  short- 

11 


C 


^C 


162 


GEOMETKY. 


est  distance  between  these  lines  is 
the  straiglit  line  PD^  which  is  at 
once  perpendicular  to  the  line  AP 
and  to  the  line  BC.  The  distance 
PD  is  the  shortest  distance  between 
them  :  because,  if  we  join  any  other 
two  points,  such  as  A  and  j5,  we 
shall  have  AB>AD,  ADyPD-, 
therefore,  still   more,  AByPD.  W'" 

Scholium  2.  The  two  lines  AE^  CB^  though  not  situated 
in  the  same  plane,  are  conceived  as  forming  a  right  angle 
with  each  other ;  because  AE  and  the  line  drawn  through 
any  one  of  its  points  parallel  to  BC^  would  make  with 
each  other  a  right  angle.  In  the  same  manner,  AB^  PD^ 
which  represent  any  two  straight  lines  not  situated  in  the 
same  plane,  are  supposed  to  form  with  each  other  the  same 
angle,  as  would  be  formed  by  AB  and  a  straight  line 
drawn  through  any  point  of  AB^  parallel  to  PD. 


PKOPOSITIOX  Yll.    THEOKEM. 

If  one  of  tico  parallel   lines    he  perpendicular    to  a  7:)?a72e,  the 
other  I'All  also  he  perpendicular  to  the  savie  j^lane. 

Let  ED,  AP,  be  two  parallel  lines;  if  AP  is  perpen- 
dicular to  the  plane  XM,  then  will  ED  be  alsc  perpendic- 
ular to  it. 

For,    through    the     parallels  A  E; 

AP,  DF,  pass  a  plane  ;   its  inter- 
section with  the  plane  MX  will  M_ 
be  PD ;   in  the  plane  3fX  draw 
BD  perpendicular  to   PD,    and 
then  draw  AD. 

Now,  BD  is  perpendicular  to 
the  plane  APDF  (p.  6,  c.)  there- 
fore, they  angle  BDE  is  a  right  angle ;  but  the  angle  EDP 
is  also  a  right  angle,  since  AP  is  perpendicular  to  PD, 
and  DF  parallel  to  AP  (b.  i.,  p.  20,  c.  1) ;  therefore,  the  line 
DE  is  perpendicular  to  the  two  straight  lines  DP,  DB; 
consequently  it  is  perpendicular  to  their  plane  3fX  (P.  4). 


M            \ 

\  "--^ 

/ 
D 

N 

BOOK    YI.  163 

Cor.  1.  Conversely  :  if  the  straight  lines  AP^  JJE^  are 
perpendicular  to  the  same  plane  MN^  they  will  be  parallel 
For,  if  they  be  not  parallel,  draw,  thiough  the  point  D^  a 
line  parallel  to  vl/,  this  parallel  will  be  perpendicular  to 
the  plane  MN ^  thereti^^e,  through  the  same  point  I)  more 
than  one  perpendicular  will  be  erected  to  the  same  plane^ 
wi^ich  is  impossible  (p.  4,  c.  2). 

Cor.  2.  Two  lines  A  and  B,  parallel  to  a  third  (7,  are 
parallel  to  each  other  ;  for,  conceive  a  plane  perpcndiculai 
to  the  line  C\  the  lines  A  and  B,  being  parallel  to  (7,  are 
perpendicular  to  this  plane  ;  therefore,  by  the  preceding 
corollary,  they  are  parallel  to  each  other. 

The  three  parallels  are  supposed  not  to  be  in  the  same 
plane ;  otherwise  the  proposition  would  be  already  proved. 
(b.  l,  p.  22). 


PROPOSITION  VIII.     THEOREM. 

If  a  straight  line   is  ^9(:rm?Ze^   to   a    line  of  a  plane,  it  w  par- 
allel to  the  ^^/a?ie. 

Let  the  straight  line  AB  be  parallel  to  the  line   CD  of 
the  plane  i\"J/;    then  will  it  be  parallel  to  the  plane  NJf. 

For,  if  the   line  AB,  which   lies  j^  j^ 

in  the  plane  ABDC,  could  meet  the 
plane  J/iVJ  it  could  only  be  in  some  M 
point  of  the  line  CJD,  the  common 
intersection  of  the  two  planes ;  but 
the  line  AB  cannot  meet  CB,  since 
they  are  parallel  (b.  i.,  d.  16) :  hence, 
it  will  not  meet  the  plane  J/iY;  therefore,  it  is  parallel  to 
that  plane  (d.  2). 

PROPOSITION  IX.     THEOREM. 

7'»w  planes   tvJiich    are  perpendicular   to  the  same  straight  lint 
are  parallel  to  each  other. 

Let  the   planes  MN,  PQ,  be    perpendicular   to   the   line 
AB,  then  will  they  be  parallel. 


16  i 


GEOMETRY. 


For,  if  tliey  can  meet  any 
where,  let  0  be  one  of  tlieir 
common  points,  and  draw  OA, 
OB.  Now,  the  line  AB,  which 
is  perpendicular  to  the  plane 
MN^  is  perpendicular  to  the 
straight  line  OA,  drawn  through 
its  foot  in  that  plane  (d.  1) ;  for 
the  same  reason  AB  is  perpendicular  to  BO  \  therefore,  there 
are  two  perpendiculars,  OA  and  OB,  let  fall  from  the  same 
point  0,  upon  the  same  straight  line,  which  is  impossible 
(b.  I.,  p.  li) ;  therefore,  the  planes  MN,  PQ^  cannot  meet 
each  other ;    consequently,  they  are  parallel. 


M 

0      "'"-f^"^ 

'-N,     L^ 

D 

p\ 

N 

\  \ 

\ 

" 

"\ 

Q 


TEOPosiTiox  X.    tiieoke:sl 


If  a  2)^cme  cut  two  parallel  plcin&s,  the  lines  of  intersection  will 

he  parallel. 

Let  the  parallel  planes  XM^  QP^  be  intersected  by  the 
plane  EH  \  then  will  the  lines  of  intersection  EF^  GH^ 
be  parallel. 

For,    if   the    lines    EF,    GH,  m_ 

lying  in  the  same  plane,  were 
not  parallel,  they  would  meet 
each  other  when  prolonged ;  and 
then  the  planes  MN^  PQ^  in  which 
those  lines  lie,  would  also  meet; 
and  hence,  the  planes  would  not 

be  parallel,  which  is  contrary  to  Q 

the  hj^pothesis. 


-^ 

F 

p 

1 

/' 

^ 

1 

\ 

/ 

G 

H 

PEOPOSITION  XL     THEOEEM. 


If  two  planes  are  parallel,  a  straight  line  ichich  is  peipendicur 
lar  to  one,  is  also  perpendicular  to  the  other. 

Let  MX,  PQ,  be  two  parallel  planes,  and  let  AB  bo 
perpendicular  to  the  plane  iVJ/;  then  will  it  also  be  per- 
pendicular to   QP, 


BOOK    YI. 


105 


For,  draw  any  line  BC  in  the 
plane  PQ,  and  through  the  lines  ^ 
AB  and  BC^  pass  a  plane  ABC^ 
intersecting  the  plane  MN  in 
AD]  the  intersection  AD  is  par- 
allel to  BG  (p.  10).  But  the  line 
AB^  being  perpendicular  to  the 
plane    i/iVJ    is    perpendicular    to  ^ 

the  straight  line  AD  (d.  1) ;  therefore,  also,  to  its  parallel 
BC  (b.  I.,  p.  20,  c.  1) ;  hence,  the  line  AB  being  perpendicu- 
lar to  any  line  BC^  drawn  through  its  foot  in  the  piano 
P§,  is  perpendicular  to  that  plane  (d.  1). 


....   li 

PX 

\  N 

\ 

\    " 

A 

PROPOSITION   XII.     THEOEEM. 


All  parallels  included  between  two  parallel  j^lanes  are  equal. 

Let  MN,  PQ,  be  two  parallel  planes,  and  IIF,  GE,  two 
parallel  lines:  then  Avill   GE—IIF. 

For,  through  the  parallels  GE, 
IIF,  draw  the  plane  EGIIF,  in- 
tersecting the  parallel  planes  in 
EF  and  GH.  The  intersections 
EF,  GH,  are  parallel  to  each  other 
(p.  10) ;  and  since  GE,  IIF  are 
parallel,  the  figure  EGIIF  is 
a  parallelogram  ;  consequently, 
EG=FH  (b.  l,  p.  28). 

Cor.  Hence,  it  follows,  that  trco  parallel  planes  are  every- 
where equidistant.  For,  suppose  EG  to  be  perpcndiculnr  to 
the  plane  PQ;  then,  the  parallel  FII  is  also  perpendicular 
to  it  (p  7),  and  the  two  parallels  are  likewise  perpendicu- 
lar to  the  plane  MN  (p.  11) ;  and  being  parallel,  they  are 
equal,  as  shown  by  the  proposition. 


i^r 

E 

>v 

p 

G/ 

/ 

/  / 

/'■■ 

\7 

II 


166  GEOMETRY. 


PEOrOSITIOX   XIII.     TIIEOKEM. 


If  two  angles,  not  situated  in  tJie  same  ^:)/cnie,  have  their  sides 
imrallel  and  lying  in  the  same  direction,  these  angles  will  he 
equal  and  their  planes  will  he  parallel. 

Let  the  angles  CAE  and  DBF,  have  the  side  A  C 
parallel  to  BD^  and  lying  in  the  same  direction  :  also,  AE 
parallel  to  BF,  and  lying  in  the  same  direction  ;  then  will 
the  angles  CAE  and  DBF  be  equal,  and  their  planes  par- 
allel. 

For,  take  AC  and  BD  equal  to 
each  other,  and  also  AE=BF; 
and  draw  CE,  BF,  AB,  CD,  EF 
Since  AC  is  equal  and  parallel  to 
BD,  the  figure  ABDC  is  a  paral- 
lelogram (b.  I.,  p.  30) ;  therefore,  CD 
is  equal  and  parallel  to  AB.  For 
a  similar  reason,  EF  is  equal  and 
parallel  to  AB;  hence,  also,  CD 
is  equal   and    parallel  to  EF  (p. 

T,  C.  2) ;    hence,  the  figure  DFEC       ^  Q 

is  a  parallelogram,  and  the  side  CE  is  equal  and  parallel 
to  DF;  therefore,  the  triangles  CAE,  DBF,  have  their  cor- 
responding sides  equal ;  consequently,  the  angle  CAE  =  DBF. 
Again,  the  plane  ACE  is  parallel  to  the  plane  BDF. 
For,  if  not,  suppose  a  plane  to  be  drawn  through  the  point 
A,  parallel  to  BDF.  If  this  plane  be  different  from  ACE, 
it  will  meet  the  lines  CD,  EF,  in  points  different  from  C 
and  E,  for  instance  in  G  and  H',  then,  the  three  lines  BA, 
DG,  FlI,  will  be  equal  (p.  12),  and  each  equal  to  AB\ 
but  the  lines  AB,  CD,  EF,  are  already  known  to  be  equal ; 
hence,  DC=DG,  and  IIF=FE,  which  is  absurd;  hen^e, 
the  plane  ACE  is  parallel  to  BDF. 

Cor.  If  two  parallel  planes  MX,  PQ,  are  met  by  two 
other  planes  CABD,  EABF,  the  angles  CAE,  DBF,  formed 
by  the  intersections  of  the  parallel  planes  are  equal ;  for, 
the  intersection  AC  is  parallel  to  BD,  and  xiE  to  BF 
(p.  10) ;  therefore,  the  angle   CAE=DBF. 


M 

J  / 

A\ 

\^' 

1^ 

4/ 

1           B 

F   / 

BOOK    YI. 


167 


PKOPOSITION  XIV.     THEOREM. 

If  three  straight  lines^  not  situated  in  the  same  plane^  are  equal 
and  parallel^  the  triangles  formed  hj  joining  the  extremities 
of  these  lines  will  he  equal,  and  their  planes  parallel. 

Let  AB^  CD^  EF^  be  three  equal  and  parallel  lines. 

Since  AB  is  equal  and  paral-  m 
lei  to  CD,  the  figure  ABDC  is 
a  parallelogram ;  hence,  the  side 
J.  (7  is  equal  and  parallel  to  BD 
(b.  l,  p.  80).  For  a  like  reason, 
the  sides  AE,  BF,  are  equal  and 
parallel,  as  also  CE,  DF  \  hence, 
the  two  triangles  A  CE,  BDF,  have 
their  sides  equal,  and  are  therefore 
equal  (b.  i.,  p.  10) ;  and  as  their  sides 
are  parallel  and  lie  in  the  same 
directions,  their  planes  are  parallel  (p.  13). 


PROPOSITION  XV.      THEOREM. 

If  two  straight  lines  he  cut  hg  three  parallel  planes,  they  will  he 
divided  proportionally. 

Suppose  the  line  xiB  to  meet  the  parallel  planes  MN, 
PQ,  Its,  at  the  points  A,  E,  B ;  and  the  line  CB  to  meet 
the  same  planes  at  the  points   C,  F,  D:    then 


AE  :  EB  :  :  CI^ 
Draw  AD  meeting  the  plane 
PQ  in  G,  and  draw  AC,  EG,  GF, 
BD.  Since  the  parallel  planes  PQ, 
RS,  are  cut  by  the  third  plane 
BAD,  the  intersections  i>i)  and  EG- 
are  parallel  (p.  10) :  and  we  have 
AE  :  EB  '.:  AG  :  GD. 
and  the  intersections  A  C,  GF, 
being  parallel, 

AG     :     GD     ::     CF    :     FD; 
hence  (b.  il,  p.  -1,  c),  AE    :     EB 


FD. 


168 


GEOMETRY. 


PKOrOSITION   XVI.     TIIEOKEM. 

If  a   line    is   jyeriiendicular    to    a  ijlane^    every  plane    passed 
through  the  i^erpendicular^  is  also  perpendicular  io  the  2^la7ie. 

Let  AP  be  perpendicular  to  the  plane  KM;    then  Avill 
every  plane  passing  through  AP  be  perpendicular  to  KJI. 

Let  BF  be  any  plane  passing  through 
AP,    and    BO  its    intersection    with    the 


B\ 


i\ 


N 


plane  MX     In  the  plane  MX,  draw  UP 

perpendicular  to  BP:    then  the  hne  AP,  ^ 

being    perpendicular   to    the    plane    MX, 

is     perpendicular    to    each    of    the    two 

straight   lines  BO,   DE.      Now,  since  AP 

and   DE  are   both   perpendicular   to    the 

common  intersection  BC,  the  angle  which 

they  form  will  measure  the  angle  between  the  planes  (d.  4) : 

but  the  angle  APD,  or  APE,  is  a  right  angle:   hence,   the 

two  planes  are  per23endicular  to  each  other. 

Scholium.  When  three  straight  lines,  such  as  AP,  BP, 
DP,  are  perpendicular  to  each  other,  any  two  may  be 
regarded  as  determining  a  plane,  and  the  three  Avill  deter- 
mine three  planes.  Now,  each  line  is  perpendicular  to  the 
plane  of  the  other  two,  and  the  three  planes  are  perpen- 
dicular to  each  other. 


PEOrOSlTION    XVIl.      TIIEOEEM. 

If  two  p)lancs  are  perpendicular  to  each  other,  a  line  drawn 
in  one  of  them  perimidicular  to  their  common  inter  sect  iorc^ 
will  he  perpendicular  to  the  other  plane. 

Let  the  plane  BA  be  perpendicular  to 
NM;  then,  if  the  line  ^P  be  perpendic- 
ular to  the  intersection  BC,  it  will  also 
be  perpendicular  to  the  plane  XM. 

For,  in  the  plane  MX,  draw  PD  per- 
pendicular to  PB ;  then,  because  the 
planes  are  perpendicular,  the  angle  APD 
is  a  right  angle  (d.  4) ;  therefore,  the  line 


BOOK    YI. 


169 


AP  is  perpendicular  to  tlie  two  straight  Hues  PB,  PD, 
passing  through  its  foot ;  therefore,  it  is  perpendicaiar  to 
their  plane  J/iY  (p.  4). 

Cor.  If  the  plane  BA  is  perpendicular  to  the  plane  J/YJ 
and  if  at  a  point  P  of  the  common  intersection  we  erect 
tt  perpendicular  to  the  plane  JLY,  that  perpendicular  will 
be  in  the  plane  BA.  For,  let  us  suppose  it  will  not,  then, 
in  the  plane  BA  draw  AP  perpendicular  to  PB,  the  com- 
mon intersection,  and  this  AP  at  the  same  time,  is  per- 
pendicular to  the  plane  MX,  by  the  theorem ;  therefore  at 
the  same  point  P  there  are  two  perpendiculars  to  the  plane 
MX,  one  out  of  the  plane  BA,  and  one  in  it,  which  is  im 
possible  (p.  4,  c.  2). 


PKOPOSITION  XVIII.     THEOREM. 

If  two  2^lcLnes  which  cut  each  other  are  peiyoidicular  to  a  third 
jplane,  their  common  irdersection  is  also  perjjendicular  to  diat 
plane. 

Let  the  planes  BA,  DA,  be  perpen- 
dicular to  NM;  then  will  their  intersec- 
tion AP  be  perpendicular  to  NM. 

For,  at  the  point  P,  erect  a  perpen- 
dicular to  the  plane  MX',  that  perpen- 
dicular must  be  at  once  in  the  plane 
AB  and  in  the  plane  AD  (p,  17,  c.) ; 
therefore,  it  is  their  common  intersection 
AP. 


PROPOSITION  XIX.     THEOREM. 

The  sum    of  either   two  of  the  plane   angles   which  include  a 
triedral  angle,  is  greater  than  the  third. 

The  proposition  requires  demonstration  only  when  the 
plane  angle,  which  is  compared  Avith  the  sum  of  the  two 
others,  is  greater  than  either  of  them.  Therefore,  suppose  the 
triedral  angle  S'  to  be  formed  by  the  three  plane  angles 
ASB,  ASO.  BSC,  and  that  the  aiu>le  ASB  is  the  greatest; 
we  are  to  show  that  AjSB<ASC-{-  USC. 


170 


GEOMETEY, 


In  tlie  plane  ASB  make  the 
angle  BSD=BSC,  and  draw  the 
straight  line  ABB  at  pleasure  ;  then 
make  ;SC=SD,  and  draw  AC,  BC. 

The  two  sides  BS,  SD,  are  equal 
to  the  two  BS,  SCj  and  the  angle 
BaSB=BSC  ;  therefore,  the  triangles 
BSD,  BSC\  are  equal  (b.  i.,  p.  5);  hence,  BD=BC. 
AB  <iAC-\-BC ;  taking  BD  from  the  one  side,  and  from  the 
other  its  equal  ^Y',  there  remains  vlZ^<^  (7.  The  two  sides 
AS,  SI),  are  equal  to  the  two  ^LS',  SC ;  the  third  side  AB 
is  less  than  the  thii-d  side  AC;  therefore,  the  angle  ASB<C 
ASC  (b.  l,  p.  9,  c.)     Adding  BSB=BSC,  we  have 

ASB+BSB,  or  ASB<ASC+BSC. 


But 


PEOPOSITIOX   XX.     THEOEEM. 

The  sum  of  the  plane  angles  which  include  any  polyedral  angle 
is  less  titan  four  rigid  angles. 

Let  S  be  the  vertex  of  a  polyedral  angle  bounded  by 
the  foccs  BSC,  CSD,  BSE,  BSa/aSB;  then  will  the  sum 
of  the  plane  angles  about  S  be  less  than  four  right  angles. 

For,  let  the  polyedral  angle  be  cut 
by  any  plane  AB,  intersecting  the  edges 
in  the  points  A,  B,  C,  B,  E,  and  the 
faces  in  the  lines  AB,  BC,  CB,  BE, 
EA.  From  any  point  of  this  plane,  as 
0,  draw  the  straight  lines  OA,  OB,  OC, 
OB,   OE. 

We  thus  form  two  sets  of  triangles, 
one  set  having  a  common  vertex  S, 
the  other  having  a  common  vertex   0, 

and  both  having  the  common  bases  AB,  BC,  CB,  BE,  EA, 
Now,  in  the  set  which  has  the  common  vertex  S,  the  sum 
of  all  the  angles  is  equal  to  the  sum  of  all  the  plane  angles 
which  comprise  the  polyedral  angle  whose  vertex  is  S,  to- 
gether with  the  sum  of  all  the  angles  at  the  bases :  viz. : 
SAB,  SB  A,  SBC,  kc. ;  and  the  entire  sum  is  equal  to  twice  as 
manv  rioht  angrles  as  there  are  triandes.     In  the  set  whose 


BOOK    YI. 


171 


common  vertex  is  (9,  the  sum  of  all  tlie  angles  is  equal 
to  the  four  right  angles  about  0,  together  with  the  inte- 
rior angles  of  the  polygon,  and  this  sum  is  equal  to 
twice  as  many  right  angles  as  there  are  triangles.  Since 
the  number  of  triangles,  in  each  set,  is  the  same,  it  fol- 
lows that  these  sums  are  equal.  But  in  the  trledral 
angle  whose  vertex  is  B,  ABS -\-SBC>ABG  (p.  19),  and 
the  like  may  be  shown  at  each  of  the  other  vertices, 
(7,  D,  E,  A  :  hence,  the  sum  of  the  angles  at  the  bases,  in 
the  triangles  whose  common  vertex  is  S^  is  greater  than 
the  sum  of  the  angles  at  the  bases,  in  the  set  whose  com- 
mon vertex  is  0 :  therefore,  the  sum  of  the  vertical  angles 
about  aS'  is  less  than  the  sum  of  the  angles  about  0 :  that 
is,  less  than  four  right  angles. 

Scholium.  This  demonstration  is  founded  on  the  suppo- 
sition that  the  polyedral  angle  is  convex,  or  that  the  plane 
of  no  one  face  produced  can  ever  meet  the  polyedral  angle; 
if  it  were  otherwise,  the  sum  of  the  plane  angles  would  no 
longer  be  limited,  and  might  be  of  any  magnitude. 

PROPOSITION  XXI.     TIIEOKEM. 


If  two  triedral  angles  are  included  by  plajie  angles  luliich  are 
et^ual  each  to  eacJi,  tJie  2^lctnes  of  the  aiual  angles  are  equally 
inclined  to  each  other. 

Let  S  and  T  be  the  vertices  of  two  triedi-al  angles,  and 
let  the  angle  ASC=I)TF,  the  angle  ASB=DTE,  and  the 
angle  BSC—ETF\  then  will  the  inclination  of  the  planes 
ASC,  ASB,  be  equal  to  that  of  the  planes  DTE,  DTE, 

For,  ha\'ing  taken  SB  at 
pleasure,  draw  BO  perpendicu- 
lar to  the  plane  ASC]  from  the 
point  0,  wdiere  the  perpendicu- 
lar meets  the  plane,  draw  0^1, 
0(7,  perpendicular  to  SA^  SO] 
draw  AB,  BC.  Next  take 
TE  =  SB]  draw  EP  perpendicular  to  the  plane  DTF\  from 
the  point  P  draw  PD^  P?\  perpendicular  respectively  to 
TD,  TE;   lastly,  draw  DE  and  EE. 


172  GEOMETRY 

The  triangle  SAB,  is  right- 
angled  at  A,  and  the  triangle 
TUB  at  B  (p.  6) :  and  since 
the  angle  A  SB = I)  TF,  we  have 
SBA=TED.  Moreover,  since 
SB  =  TE,  the  triangle  SAB  is 
equal  to  the  triangle  TDE\ 
therefore,  SA=-TD,  and  AB=DE 

In  like  manner,  it  may  be  shovrn,  that  SC=JE,  and 
BC=EF.  That  proved,  the  quadrilateral  ASCO  is  equal 
to  the  quadrilateral  i^Ji^P:  for,  place  the  angle  J6'6^  upon 
its  equal  DTF -,  because  SA=TD,  and  SC=TF,  the  point 
A  will  foil  on  B,  and  the  point  C  on  F;  and,  at  the  same 
time,  AO,  which  is  perpendicular  to  SA,  will  fall  on  DP, 
which  is  perpendicular  to  TB,  and,  in  like  manner,  OC  on 
PF\  Avherefore,  the  point  0  will  fall  on  the  point  P,  and 
hence,  AG  is,  equal  to  DP. 

But  the  triangles  A  OB,  DPE,  are  right-angled  at  0  and 
P;  the  hypothenuse  AB=DE,  and  the  side  AO==DP: 
hence,  those  triangles  are  equal  (b.  i.,  p.  17) ;  and,  conse- 
quently, the  angle  OAB  =  PDE.  But  the  angle  OAB  mea- 
sures the  inclination  of  the  two  faces  ASB,  A  SO;  and  the 
angle  PDF  measures  tliat  of  the  two  faces  DTE,  DTE  \ 
hence,  those  two  inclinations  are  equal  to  each  other. 

Scholium  1.  It  must,  however,  be  observed,  that  the 
angle  A  of  the  right-angled  triangle  AOB  is  properly  the 
inclination  of  the  two  planes  ASB^  ASC,  on\Y  when  the 
perpendicular  BO  falls  on  the  same  side  of  SA,  with  SC ; 
for,  if  it  fell  on  the  other  side,  the  angle  of  the  two  planes 
would  be  obtuse,  and  the  obtuse  angle  together  with  the 
angle  A  of  the  triangle  OAB  would  make  two  right  angles. 
But  in  the  same  case,  the  angle  of  the  two  planes  TDE^ 
TDF,  would  also  be  obtuse,  and  the  obtuse  angle  together 
with  the  angle  D  of  the  triangle  DPF,  would  make  two 
right  angles ;  and  the  angle  A  being  thus  alwa^'s  equal  to 
the  angle  D,  it  would  follow  that  the  inclination  of  the 
two  planes  ASB,  ASC,  must  be  equal  to  that  of  the  two 
planes  TDF,   TDF. 

Scholium  2.    If  two  triedral  angles  are  included  by  three 


BOOK    VI.  173 

plane  angles,  respectively  equal  to  each  other,  and  if,  at 
the  same  time,  the  equal  or  homologous  angles  are  disposed 
in  the  same  order^  the  two  triedral  angles  will  coincide  when 
applied  the  one  to  the   other,  and   consequently,  are  equal 

(A.  14). 

For,  we  have  already  seen  that  the  quadrilateral  SAOC 
may  be  placed  upon  its  equal  TDPF  \  thus,  placing  8A  upon 
TD^  SO  falls  upon  TF^  and  the  point  0  upon  the  point  P. 
Bat  because  the  triangles  AOB,  1)PE,  are  equal,  Oi>,  per- 
pendicular to  the  plane  ASC^  is  equal  to  PE,  perpendicu- 
lar to  the  plane  TDF  \  besides,  these  perpendiculars  lie  in 
the  same  direction  ;  therefore,  the  point  B  will  fall  upon 
the  point  E^  the  line  SB  upon  TE,  and  the  two  angles  will 
wholly  coincide. 

Scholium  3.  The  equality  of  the  triedral  angles  does 
not  exist,  unless  the  equal  faces  are  arranged  in  the  same 
manner.  For,  if  they  were  arranged  in  an  inverse  order^  or, 
what  is  the  same,  if  the  perpendiculars  OB,  PE^  instead  of 
lying  in  the  same  direction  v/ith  regard  to  the  planes  ASC^ 
DTF^  lay  in  opposite  directions,  then  it  would  be  impossi- 
ble to  make  these  triedral  angles  coincide  the  one  with  the 
other.  The  theorem  would  not,  however,  on  this  account, 
be  less  true,  viz. :  that  the  faces  containing  the  equal 
angles  must  be  equally  inclined  to  each  other  ;  so  that 
the  two  triedral  angles  Avould  be  equal  in  all  their  con- 
stituent parts,  without,  however,  admitting  of  superposi- 
tion. This  sort  of  equality,  which  is  not  absolute,  or 
such  as  admits  of  superposition,  ought  to  be  distinguish- 
ed by  a  particular  name :  we  shall  call  it,  equaUty  hy  sym- 
metry. 

Thus,  those  two  triedral  angles,  which  are  formed  by 
faces  respectively  equal  to  each  other,  but  disposed  in  an 
inverse  order,  will  be  called  triedral  angles  equal  hy  symme- 
try ^  or  simply  symmetrical  angles. 


BOOK    VII 


P  0  L  Y  E  D  R  0  N  S 


DEFIXITIONS, 


1.  PoLTEDROX  is  a  name  given  to  any  solid  bcunded 
by  polygons.  The  bounding  polygons  are  called  faces  of 
the  polyedron  ;  and  the  straight  line  in  ^Yhich  any  two 
adjacent  faces  meet  each  other,  is  called  an  edge  of  the 
polyedron. 


2.  A  Prism  is  a  poh^edron  in  whicli  two 
of  the  faces  are  equal  polj-gons  with  their 
planes  and  homologous  sides  parallel,  and  all 
the  other  faces  parallelograms. 


3.  The  equal  and  parallel  polygons  are  called  hases  of 
the  pri?m — the  one  the  lower,  the  other,  the  upper  base — 
and  the  parallelograms  taken  together,  make  up  the  lateral 
•JT  convex  surface  of  the  prism. 

4.  The  Altitude  of  a  prism  is  the  distance  betwzin 
its  two  bases,  and  is  measured  by  a  line  drawn  froi.;  a 
point  in  one  base,  perpendicular  to  the  plane  of  the  oti    f. 

5.  A  right  prism  is  one  whose  edges, 
formed  by  the  intersection  of  the  lateral  faces, 
are  perpendicular  to  the  planes  of  the  bases. 
Each  ed,2fe  is  then  equal  to  the  altitude  of  the 
prism.  In  every  other  case,  the  prism  is 
vUiqiie^  and  each  edge  is  then  greater  than  the 
altitude. 


r 


BOOK  YII. 


175 


6.  A  Triangular  Prism  is  one  whose  bases  are  tri- 
angles :  a  quadrangular  prism  is  one  whose  bases  are  quad- 
rilaterals :  a  pentangular  prkm  is  one  whose  bases  are  pen- 
tagons: a  hexangular  prism  is  one  whose  bases  are  hexa- 
gon's, ko,. 

7.  A  Parallelopipedon"  is  a  prism  whose  bases  are 
parallelograms. 


8.  A  Rectangular  Parallelopipe- 
DON  is  one  whose  faces  are  all  rectangles. 
When  the  flices  are  squares,  it  is  called 
a  cube^  or  regular  hexaedron. 


9.  A  Pyramid  is  a  solid  bounded  by 
a  polygon,  and  by  triangles  meeting  at  a 
common  point,  called  the  vertex.  The 
polygon  is  called  the  base  of  the  pyra- 
mid, and  the  triangles,  taken  together, 
the  convex^  or  lateral  surface.  The  pyra- 
mid, like  the  prism,  takes  different  names, 
according  to  the  form  of  its  base  :  thus, 
it  may  be  triangular,  quadrangular,  pent- 
angular, &c. 

10.  The  Altitude  of  a  pyramid  is  the  perpendicular 
let  fall  from  the  vertex  on  the  plane  of  the  base. 

11.  A  PtTGiiT  PYRA:\riD  is  one  whose  base  is  a  regular 
polygon,  and  in  which  the  perpendicular  let  fall  from  the 
vertex  upon  the  base  passes  through  the  centre  of  the  base. 
This  perpendicular  is  then  called  the  axis  of  the   pyramid. 

12.  The  Slant  Height  of  a  right  pyramid,  is  the  per- 
pendicular let  fall  from  the  vertex  to  either  side  of  the 
polj^gon  which  forms  the  base. 


13.  If  a  pyramid  is  cut  by  a  plane 
parallel  to  its  base,  forming  a  second 
base,  the  part  lying  between  the  bases, 
is  called  a  truncated  pyramid^  or  frustum 
of  a  pyramid. 


17(3  GEOMETRY. 

1-i.  Tlie  altitude  of  a  frustum  is  the  perpendicular  dis- 
tance between  its  bases :  and  the  slant  height^  is  that  por- 
tion of  the  slant  height  of  the  j^jramid  intercepted  between 
tl^e  bases  of  the  frustum. 

15. ,  The  diagonal  of  a  polyedron  is  a  line  joining  the 
vertices  of  an}-  two  of  its  angles,  not  in  the  same  face. 

16.  Similar  pohjedrons  are  those  whose  polyedral  angles 
are  equal,  each  to  each,  and  which  are  bounded  by  the 
same  number  of  similar  fliccs. 

17.  Parts  which  are  like  placed,  in  similar  poljedrons, 
v.'hether  faces,  edges,  or  angles,  are  called  homologous. 

18.  A  regular  polyedron  is  one  whose  faces  are  equal 
and  regular  polygons,  and  whose  polj^edi^al  angles  are  equal. 

PEOPOSITION  I.     TIIEOKEM. 

The  convex  surface  of  a   rigid  prism  is  equal  to  the  perimeter 
of  either  hase  multiplied  hy  its  altitude. 

Let  ABCDE-K  be  a  right  prism :  then  will  its  convex 
surface  be  equal  to 

{AB-\-BC+CD^-DE-\-EA)xAF. 

For,  the  convex  surface  is  equal  to 
the  sum  of  all  the  rectangles  AG^  BH^ 
CI,  Dl\,  EF,  which  compose  it.  Now,  K 
the  altitudes  AF,  BG,  CII,  &c.,  of  the 
rectangles,  are  equal  to  the  altitude  of 
the  prism,  and  the  area  of  each  rect- 
angle is  equal  to  its  base  multiplied  by  -^ 
its  altitude  (b.  iv.,  p.  5).  Hence,  the  sum 
of  these  rectangles,  or  the  convex  sur- 
foce  of  the  prism,  is  equal  to 

{AB+BC+CD+DE-\-EA)xAF- 
that  is,  to   the   perimeter  of  the  base  of  the  prism  multi- 
plied by  the  altitude. 

Cor.  If  two  right  prisms  have  the  same  altitude,  their 
convex  surfaces  are  to  each  other  as  the  perimeters  of 
their  bases. 


BOOK   yii. 


177 


TKOrOSITION  II.     TIIEOEEM. 


In   every   prism,   the   sections  formed   hy  parallel  planes,    are 
equal  polygons. 


Let  tlie  prism  All  be  intersected  by  tlie  parallel  planes 
NP,  SV]  then  are  the  polygons  KOPQR,  STVXY,  equal. 

For,  the  sides  ST,  NO,  are  parallel, 
being  the  intersections  of  two  parallel 
planes  with  a  third  plane  ABGF;  these 
same  sides,  ST,  NO,  are  included  be- 
tween the  parallels  NS,  OT,  which  are 
edges  of  the  prism  :  hence,  NO  is 
equal  to  ST.  For  like  reasons,  the 
sides  OP,  PQ,  QR,  &c.,  of  the  section 
NOPQB,  are  equal  to  the  sides  7'V, 
VX,  XT,  &c.,  of  the  section  STVXY, 
each  to  each  ;  and  since  the  equal 
sides  are  at  the  same  time  j)arallel,  it 
follows  that  the  angles  NOP,  OPQ,  &c.,  of  the  first  section, 
are  equal  to  the  angles  STV,  TVX,  &c.,  of  the  second,  each 
to  each  (b.  vi.,  P.  13).  Hence,  the  two  sections  NOPQB, 
STVXY,  are  equal  polygons. 

Cor.   Every  section  of  a  prism,  parallel  to  the  bases,  is 
equal  to  either  base. 


PROPOSITION  III.     THEOREM. 

Tf  a  pyramid  he  cut  hy  a  plane  parallel  to  its  hase: 

1st.    The  edges  and  the  altitude  ivill  he  divided  2)roportionaUi/ : 

2d.    The  section  will  he  a  polygon  similar  to  the  hase. 


Let  the  pyramid  S- ABODE,  of  which  SO  is  the  altitude 
be  cut  by  the  plane  ahcde-,    then  will 

Sa    :     SA     ::     So    :     SO, 

and  the  same  for  the  other  edges ;    and  the  polygon  ahcde^ 
will  be  similar  to  the  base  ABODE. 

12 


178 


GEOMETRY. 


Fir^f.  Since  the  planes  ABC,  ahc,  are 
parallel,  their  intersections  J  i>,  ah,  by  the 
third  ])hine  SAB,  are  also  parallel  (b.  VI., 
P.  10) ;  hence,  the  triangles  SAB,  Scib,  are 
similar  (b.  iv.,  p.  21),  and  we  have 

>S;i     :     Sa     :  :     SB    :     Sb ; 
for  a  like  reason,  we  have 

SB    :     Sb     :  :     SO    :     Sc : 
and  so  on.     Hence,  the  edges  SA,  SB,  SC^ 
&c.,  are    cnt    proportionally  in  a,  h,  r,  &c. 
The  akitude  SO  is  likewise  cnt  in  the   same  proportion,  at 
the  point  0 ;    for  BO  and  bo  are  parallel,  therefore,  we  have 

SO     :     So     ::     SB    :     Sb. 

Secondhj.  Since  ab  is  parallel  to  AB,  be  to  BC,  cd  to  CD^ 
kc,  the  angle  abc  is  eqnal  to  ABC,  the  angle  bed  to  BCD, 
and  so  on  (b.  vi.,  p.  13).  Also,  by  reason  of  the  simikir 
triangles  SAB,  Sab,  we  have 

AB    :     ab     ::     SB    :     Sb; 
and  by  reason  of  the  similar  triangles  SBC,  Sbe,  we  have 

SB    :     Sb     ::     BC    :     be, 

hence,  AB    :     ab     :  :     BC    :     be; 

we  might  likewise  have 

BC    :     be     ::     CD     :     cd, 

and  so  on.  Hence,  the  polygons  ABODE,  abcde  have  their 
angles  e<[nal,  each  to  each,  and  their  sides,  taken  in  the 
same  order,  proportional ;  hence,  they  are  similar  (b.  iv.,  D.  1). 

Cor.  1.  Let  SABCDE, 
S-XYZ,  be  two  pyramids, 
having  a  common  vertex 
and  their  bases  in  the  same 
plane  ;  if  these  pyramids  are 
cut  by  a  plane  parallel  to 
thQ  plane  of  their  bases,  the 
sections,  abcde,  xijz,  will  he  to 
each  other  as  the  bases  ABCDE, 
XYZ. 


BOOK    VII, 


179 


For,  the  polygons  ABCDE^  ahccle^  being  similar,  their 
surfaces  are  as  the  squares  of  the  homologous  sides  AB^ 
ah ;  that  is,  B.  iv.,  P.  27), 

ABODE    :     alcde     :  :     ^07'     :     ab'. 
but,  AB    :     ah     '.:     SA     :     Sa  ; 

hence,  ABODE    :     ahcde     :  :     SA"     :     Sa , 

For  the  same  reason, 

XYZ    :     xyz     :  :     SX      :     Sx . 
But  since  ahc  and  x^jz  are  in  one  j)lane,  we  have  likewise 
(b.  VI.,  P.  15), 

SA     :     Sa     :  :     SX    :     Sx ; 
hence,         ABODE    :     ahcde     :  :     XFZ    :     xi/z  ; 
therefore,  the    sections  ahcde^  xi/z,  are   to    each  other  as  the 
bases  ABODE,  XYZ. 

Cor.  2.  If  the  bases  ABODE^  XYZ,  are  equivalent,  any 
sections  ahcde,  xyz,  made  at  equal  distances  from  the  bases, 
are  also  equivalent. 

PPvOPOSITION   IV.      TIIEOKEM. 


The  convex  surface  of  a  right  pyramid  is  equcd  to  the  2)crimeter 
of  its  base  multiplied  by  Italf  the  slant  height. 

Let  aS'  be  the  vertex,  ABODE  the  base,  and  SF  the  slant 
height  of  a  right  pj-ramid ;  then  the  convex  surface  is  equal 
to  l.SFx^AB  +  BO+OD  +  DE+EA). 

For,  since  the  pyramid  is  right,  the 
point  0,  iu  which  the  axis  meets  the 
base,  is  the  centre  of  the  polygon 
A  BODE  (D.  11) ;  hence,  the  lines  OA, 
OB,  00,  &c.,  drawn  to  the  vertices  of 
the  base,  are  equal. 

In  the  right-angled  triangles  SAO, 
SBO,  the  bases  and  perpendiculars  are 
c<iual :  hence,  the  hj^pothenuses  are 
equal:  and  it  may  be  proved  in  the 
same  way,  that    all    the    edges    of  the    right    pyramid   are 


180  GEOMETRY. 

equal.  The  triangles,  tlierefore,  which 
form  the  convex  surface  of  the  prism 
are  all  equal  to  each  other.  But  the 
area  of  either  of  these  triangles,  as  USA, 
is  equal  to  its  base  IJA,  multiplied  by 
half  the  perpendicular  aSF,  which  is  the 
•slant  height  of  the  pyramid :  hence,  the 
area  of  all  the  triangles,  or  the  convex 
surface  of  the  pyramid,  is  equal  to  the 
perimeter  of  the  base  multiplied  by  half 
th3  slant  height. 

Cor.  The  convex  surface  of  the  frustum  of  a  right  j>yrcimid 
is  equal  to  half  the  sum  of  the  perimeters  of  its  upper  and 
lower  bases  multiplied  hy  its  slant  height. 

For,  since  the  section  ahccle  is  similar  to  the  base  (p.  3), 
and  since  the  base  ABODE  is  a  regular  polj^gon  (d.  11),  it 
follows  that  the  sides  ea,  ah,  he,  cd,  and  de,  are  all  equal  to 
each  other.  Hence,  the  convex  surface  of  the  frustum 
ABCDE-d  is  composed  of  the  equal  trapezoids  EAae,  ABha, 
kc,  and  the  perpendicular  distance  between  the  parallel 
sides  of  either  of  these  trapezoids  is  equal  to  Ff  the  slant 
height  of  the  frustum.  But  the  area  of  either  of  the  trap- 
ezoids, as  AEea,  is  equal  to  i{EA-hea)xFf  (b.  iy.,  p.  7): 
hence,  the  area  of  all  of  them,  or  the  convex  surface  of 
the  frustum,  is  equal  to  half  the  sum  of  the  perimeters  of 
the  upper  and  lower  bases  multiplied  by  the  slant  height. 


PEOPOSITION    V.     THEOEEM. 

If  the  three  faces  ichich  include  a  triedrcd  angle  of  a  prism  are 
equcd  to  the  three  faces  ichich  include  a  triedral  angle  of  a 
second  p)risni,  each  to  each,  and  are  like  placed,  the  two 
prisms  are  equal. 

Let  B  and  h  be  the  vertices  of  two  triedral  angles  in 
eluded  by  faces  respectively  equal  to  each  other,  and 
similarly  placed ;  then  will  the  prism  ABCDE-K  be  equal 
to  the  prism  abcde-Jc. 

For,  place  the  base  ahcde  upon  the  equal  base  ABODE; 


181 


then,  since  the  triedral  angles  at  h  and  B  are  equal,  tho 
parallelogram  Ih  will  coincide  with  i>//,  and  the  parallelo- 
gram hf  w-ith  BF.  But  the  two  upper  bases  being  equal 
to  their  corresponding  lower  bases,  are  equal  to  each  other, 
and  consequently,  will  coincide  :  hence,  hi  Avill  coincide 
with  in^  ik  with  IK^  Icf  with  KF  ]  and  therefore,  tho 
lateral  faces  of  the  prisms  will  coincide  :  hence,  the  two 
prisms  coinciding  throughout,  are  equal  (a.  1^). 

Cor.  Two  right  prisms,  luhich  have  equal  bases  and  equal 
altitudes^  are  equal.  For,  since  the  side  AB  is  equal  to  a/;, 
and  the  altitude  BG  to  Ig^  the  rectangle  ABGF  is  equal  to 
ahgf-^  so  also,  the  rectangle  BGHG  is  equal  to  Ighc,  and 
thus  the  three  faces,  which  include  the  triedral  angle  B^ 
are  equal  to  the  three  which  include  the  triedral  angle  Z>, 
each  to  each.     Hence,  the  two  prisms  are  equal. 


PROPOSITION    VI.     THEOREM. 


In    every   parallelopipedon,    the    opposite  faces    are    equal    aiid 

jyarallel. 


E 


II 


Let  A  BCD  be  a  parallelopipedon,  then  Avill  its  opposite 
faces  be  equal  and  parallel. 

For,  the  bases  ABCD,  EFGTI,  are 
equal  parallelograms,  and  have  their 
planes  parallel  (d.  7).  It  remains  oul}^ 
to  show,  that  the  same  is  true  of  any 
two  opposite  lateral  faces,  such  as 
BCGF,  ADim 


182  GEOMETRY. 

IvTow,  BC  is  equal  and  parallel  to 
AD,  because  the  base  ABCI)  is  a  par- 
allelogram ;  and  since  the  lateral  faces 
are  also  parallelograms,  BF  is  equal  and 
parallel  to  AE^  and  the  like  niay  be 
chown  for  the  sides  FG  and  EII^  CG  and 
DII]  hence,  the  angle  CBF  is  equal  to  the  angle  DAE^  and 
the  planes  DAE^  CBF^  are  parallel  (b.  vi.,  r.  13);  and  the 
parallelogram  BCGF,  is  equal  to  the  parallelogram  ABIIE. 
In  the  same  way,  it  may  be  shown  that  the  o])posite  paral- 
lelogTams  ABFE,  DCGH^  are  equal  and  parallel. 

Cor.  1.  Since  the  parallelopipedon  is  a  solid  bounded 
by  six  faces,  of  Avhich  any  two  lying  opposite  to  each 
other,  are  ec[ual  and  parallel,  it  follows  that  any  face  and 
the  one  opposite  to  it,  rnay  be  assumed  as  the  bases  of  the 
parallelopipedon. 

Cor.  2.  The  diagonals  of  a  paixiUelopijoedon  Used  each  other. 
For,  suppose  two  diagonals  BH^  BF^  to  be  drawn  through 
opposite  vertices.  Draw  also  BD,  FII.  Then,  since  BF  is 
equal  and  parallel  to  DTI,  the  figure  BDIIF  is  a  parallelo- 
gram ;  hence,  the  diagonals  BH^  DF^ 
mutually  bisect  each  other  at  E  (b.  I.,  P. 
81).  In  like  manner,  it  may  be 
shown  that  the  diagonal  BII  and  any 
other  diagonal  bisect  each  other  ;  hence, 
the  four  diagonals  mutually  bisect  each 
other,  in  a  common  point.  If  the  six 
faces  are  equal  to  each  other,  this  point  may  be  regarded 
as  the  centre  of  the  parallelopipedon. 

Scholium.  If  three  straight  lines  AB,  AE^  AD,  passing 
through  the  same  point  ^1,  and  making  given  angles  with 
each  other,  are  known,  a  parallelopipedon  may  be  formed 
on  these  lines.  For  this  purpose,  conceive  a  plane  to  be 
passed  through  the  extremity  of  each  line,  and  parallel  to 
the  plane  of  the  other  two,  that  is,  through  the  point  B 
pass  a  plane  parallel  to  DAE,  through  D  a  plane  parallel 
to  BAE,  and  through  E  a  plane  parallel  to  BAD.  The 
mutual  intersections  of  these  planes  will  form  the  edges  of 
the  parallelopipedon  required. 


BOOK    VII. 


18:i 


PKOPOSITION  VII.    TIIEOKEM. 

If  a  plane  he  passed,  through  the  opposite  diagonal  edges  of  a 
parallelopipedon^  it  luill  divide  the  solid  into  two  C(piivalent 
triangular  prisms. 

Let  the  parallelopipedon  ABCD-H  be  divided  by  the 
plane  BDHF^  passing  through  the  opposite  edges  Bt\  DH: 
then  will  the  triangular  prism  ABI)-II,  be  equivalent  to 
the  triangular  prism  i>  CD-//. 

For,  through  the  vertices  B  and  F^ 
pass  the  planes  Bcda,  Fghe^  at  right 
Jingles  .to  the  edge  BF,  the  former  cut- 
ting the  three  other  edges  of  the  par- 
allelopipedon prolonged  in  the  points 
c,  c/,  a,  the  latter  in  the  points  g^  h,  e. 

Now,  the  sections  Beda,  Fghe,  are 
equal  parallelograms.  For,  the  cutting 
planes  being  perpendicular  to  the  same 
straight  line  BF,  are  parallel  (b.  vi.,  p. 
9) :  hence,  the  sections  are  equal  (p.  2) ;  and  they  are  par- 
allelograms because  Ba,  cd,  two  opposite  sides  of  the  same 
section,  are  formed  by  the  meeting  of  a  plane  aBcd,  with 
two  parallel  planes  ABFF,  DCGH  (b.  vi.,  p.  10).  For  a 
similar  reason  Be  and  ad  are  parallel ;  hence,  the  figures 
are  equal  parallelograms. 

For  a  like  reason  the  figure  aBFe  is  a  parallelogram; 
so  also,  are  BcgF,  cghd,  adlie,  the  other  lateral  flxccs  of  the 
solid  aBcd-h ;  hence,  that  solid  is  a  prism  (d.  2),  and  that 
prism  is  right,  since  the  edge  BF  is  perpendicular  to  its 
bases. 

But  the  right  prism  aBcd-h  is  divided  by  the  plane  BE 
into  two  equal  right  prisms  aBd-h,  Bcd-h ;  for,  the  bases 
aBd,  Bed,  are  equal,  being  halves  of  the  same  parallelo- 
gram, and  since  the  prisms  have  the  common  altitude  BF, 
they  are  equal  (p.  5,  c.) 

It  is  now  to  be  proved  that  the  oblique  triangular 
prism  ABD-H  is  equivalent  to  the  right  triangular  prism 
aBd-li,  Since  these  prisms  have  a  common  part  ABD-h,  it 
will  only  be  necessary  to  prove    that   the   remaining  parts, 


184 


GEOMETKY. 


namely,  the  solids  aBd-D,  eFh-H^  are 
equivalent.  Since  ABFF,  aBFe,  are 
parallelograms,  the  sides  AB,  ae,  are 
each  equal  to  BF ;  hence,  they  are 
equal  to  each  other;  and  taking  away 
the  common  part  J.e,  there  remains 
Aa=Fe.  In  the  same  manner  it  may 
be  proved  that  Dd=ffh. 

To  bring  about  the  superposition  of 
the  two  solids,  eFh-H,  aBd-D,  let  the 
base  eFh  be  placed  on  the  equal  base  aBd — the  point  c 
falling  on  «,  the  point  h  on  d :  the  edges  eE^  liH,  will  then 
coincide  with  aA,  dD^  since  all  the  edges  are  perpendicular 
to  the  same  plane  aBcd.  Hence,  the  two  solids  will  coin- 
cide exactly  with  each  other;  consequently,  the  oblique 
prism  ABD-H  is  equivalent  to  the  right  prism  aBd-h.  In 
the  same  manner,  it  may  be  shown  that  the  oblique  prism 
BCD-II  is  equivalent  to  the  right  prism  Bcd-h.  But  the 
two  right  prisms  have  been  proved  equal:  hence,  the  two 
triangular  prisms  ABD-H^  BCD-H^  being  equivalent  Xo 
equal  right  prisms,  are  equivalent  to  each  other. 

Cor.  Every  triangTdar  prism  ABDH  is  half  the  paral- 
lelopipedon  J.6^,  ha^-ing  the  same  triedral  angiv.  A,  and  the 
same  edses  AB^  AD^  AE. 


rPwOPOSITIOX   VIII.     THEOKEM. 


If  two  parallehpipedons  have  a  common  loicer  base,  and  their 
tipper  bases  in  the  same  plane  and  between  the  same  p)ciral' 
lels^  they  are  equivalent. 

Let  the  parallelopipedons  AG^  AL^  have  the  common 
case  ABCD,  and  their  upper  bases  EG^  IL^  in  the  same 
plane,  and  between  the  same  parallels  EK^  HL ;  then  will 
they  be  equivalent. 

There  may  be  three  cases,  according  as  EI  is  greatei 
than,  equal  to,  or  less  than  EF\  but  the  demonstration,  for 
each  case,  is  the  same. 

TTe  will  show,  in  the  first  place,  that  the  triangular 
nrisms  AIE-H,  BKF-G  are  eciual.     Since  EF  and  IK    are 


't 


BOOK    VII.  185 

eacTi    equal   to  AB  (b.  i.,  n GM L 

p.  28),  they  are   equal   to 
eacli  otlier.      Add  FI  to 
eacli,  and  we  have 
FI=FK: 
and  since  the  angle  AEF      'D\ 
is    equal   to    BFK    (b.  I., 
P.  20,  c.  3) ;    the    triangle  ^  ^ 

AEI  is  equal  to  the  triangle  BFK  (b.  I.,  P.  5).  Again, 
since  EI  is  equal  to  FK^  and  EH  equal  and  parallel  to 
FG,  the  parallelogram  EM  is  equal  to  the  ])arallelog]'am 
FL  (b.  I.,  p.  28,  c.  2) :  also,  the  parallelogram  AH  is  equal 
to  the  parallelogram  CF  (p.  6)  :  hence,  the  three  faces 
which  include  the  polycdrnl  angle  at  E  are  respectively 
equal  to  the  three  which  include  the  polyedral  angle  at  F^ 
and  being  like  placed,  the  ti"iangular  prism  AIE-H  is  equal 
to  the  triangular  prism  BKF-G  (p.  5). 

But,  if  the  triangular  prism  AIE-H  be  taken  away 
from  the  solid  AL^  there  will  remain  the  parallelopipedon 
ABCD-M]  and  if  the  equal  triangular  prism  BKF-G  be 
taken  away  from  the  same  solid,  there  will  remain  the 
parallelopipedon  ABCB-H ;  hence,  the  two  parallelopipcdons 
ABCD-M,  ABCB-H,  are  equivalent. 

PKOPOSITION  IX.     THEOREM. 

Two  parallelopipedons^  having  their  loiver  bases  equal,  and  equal 
altitudes^  are  equivalent. 

Let  the  parallelopipcdons  AG^  AL,  have  the  common 
base  A  BCD,  and  equal  altitudes;  then  Avill  their  upper 
bases,  EFGH,  IKL3f,  be  in  the  same  plane ;  and  the  tvro 
parallelopipcdons  will  be  equivalent. 

For,  let  the  edges  FE,  GH,  be  prolonged,  as  also,  XL 
and  IM,  till,  by  their  intersections,  they  form  the  paral- 
lelogram NOPQ,  in  the  plane  of  the  upper  bases :  this 
parallelogram  will  be  equal  to  either  of  the  bases  IL,  EG, 
For,  the  upper  bases  IL,  EG,  being  each  equal  to  the 
common  base  siO,  are  equal  to  each  other.  But  OP 
which   is    equal    to    FG^  is  also    equal  to  KL,  and    OX  is 


186 


GEOMETEY, 


equal  to  AT,  being  be 
tween  the  same  paral- 
lels :  hence,  the  paral- 
lelogram XP  is  equal 
to  JL  or  FG  (b.  I.,  r. 
28,  c.  2). 

Now,  if  a  third  par- 
allelopipeJon  be  con- 
ceived, having  for  its 
iQwer  base  the  paral- 
lelogram ABCD,  and  for 
its  ujiper  base  NOPQ^ 
this  thii'd  parallelopipedon  will  be  equivalent  to  the  paral- 
lelopipedon  AG^  since  they  have  the  same  lower  base,  and 
their  u])per  bases  lie  in  the  same  plane  and  between  the 
Bame  paraltels,  QG,  NF  (p.  8).  For  a  like  reason,  this  third 
parallelopipedon  will  also  be  equivalent  to  the  parallelo- 
pipedon AL ;  hence,  the  two  parallelopipedons  A  6\  AL, 
which  have  equal  bases  and  equal  altitudes,  are  equivalent. 


riiorosiTiox  x.    theorem. 

Any  pnralMopipedon  may  he  dtanrjed  into  an  equivalent  reel- 
anynlar  jiarallelopijxdon  having/  an  eijual  altlttale  and  an 
equicalent  hase. 

Let  ABCD-II  be  any  parallelopipedon. 

From  the  vertices  A, 
B,  C  ]),  draw  A  I,  BK, 
CL^  DM^  perpendicular  to 
the  plane  of  the  lower 
base,  and  equal  to  the 
altitude  of  A  G :  there  will 
thus  be  formed  the  paral- 
lelopipedon AL  equiva- 
lent to  ^6^  (p.  9),  and 
having  its  lateral  faces 
AIv,  BL,  kc,  rectangles. 
Now,  if  the  base  ABCB 
is  a  rectangle,  AL  will    be    a   rectangular   parallelopipedon 


BOOK  YII. 


187 


D 


LP 


A" 


\i^ 


-In 


equivalent  to  AG,  and    consequently,  the    parallelopipedou 
required. 

But  if  ABCD  is  not  a  rectangle,  draw      M  Q 
AO  and    BN  perpendicular    to    JJC,  and 
OQ  and  NP  perpendicular   to    the  base;  i  I 

we  shall  then  have,  a  rectangular  piral- 
lelopipedon  JLi>i\'6>-() :  for,  by  construc- 
tion, the  bases  ABXO,  and  IKPQ,  are 
rectangles ;  so  also,  are  the  lateral  faces, 
the  edges  AT,  OQ,  &c.,  being  perpendicu- 
lar to  the  plane  of  the  base  ;  hence,  the 
solid  AP  is  a  rectangular  parallellopipedon.  But  the  two 
parallelopipedons  AP,  AL,  may  be  conceived  as  having  the 
same  base  ABKI,  and  the  same  altitude  ^4  0 :  hence,  the 
parallelopipedon  AG,  which  was  at.  first  changed  into  an 
equivalent  parallelopipedon  AL,  is  now  changed  into  an 
equivalent  rectangular  parallelopipedon  AP,  having  the 
same  altitude  Al,  and  a  base  ABXO  equivalent  to  the  base 
ABCD. 


*B 


PROPOSITION   XI.     THEOREM. 

Two  rectangular  j^'^'^^^^^^f^^opipedons,  ichich  have   equal  lascs,  are 
to  each  other  as  their  altitudes. 


E 


B 


n 


-1— G 


Let  the  parallelopipedons  AG,  AL,  have  the  common 
base  BD,  then  will  they  be  to  each  other  as  their  altitudes 
AL]  AL 

First.  Suppose  the  altitudes  AE,  AL,  to 
be  to  each  other  as  two  whole  numbers, 
as  15  is  to  8,  for  example.  Divide  AE 
into  15  equal  parts,  whereof  AL  will  con- 
tain 8 ;  and  through  x,  y,  z,  &c.,  the  points 
of  division,  pass  planes  j)arallel  to  the 
common  base.  These  planes  will  divide 
the  solid  A  G  into  15  parallelopipedons,  all 
equal  to  each  other,  because  they  have 
equal  bases  and  equal  altitudes — equal 
bases,  since  every  section  KLML,  parallel 
to  the  base  ABCD,  is  equal  to  that   base  (r.  2),  equal  alti- 


l\ 

F 

K 

0 

I 

■m 

\ 

\ 

r 

M 

z 

1^ 

j 

i 

A 

\ 

1) 

\ 

^L 


188 


GEOMETRY, 


tudes,  because  the  altitudes  are  tlie  equal 
divisions,  Ax,  xy,  yz^  ko,.  But  of  tliose  15 
equal  parallelopipedons,  8  are  contained 
in  J.Z;  hence,  the  solid  AG  is  to  the 
solid  AL  as  15  is  to  8,  or  generally,  as 
the  altitude  AE  is  to  the  altitude  AL 

Second.  If  the  ratio  of  AE  to  AI 
cannot  be  expressed  exactly  in  numbers, 
it  may  still  be  shown,  that  we  shall 
have 


E 

Tf 

\ 

F 

\ 

0 

. 

1 
1 

I 

-m 

\ 

M 

\ 

z 

K 

y-] 

a- 

A 

\ 
\ 

\ 

B" 


solid  AG    :     solid  AL    ::     AE    :     AL 
For,  if  this  proportion  is  not  correct,  suppose  we  have, 
sol.  AG    :     sol.  AL     :  :     AE    :     AO  greater  than  AL 

Divide  AE  into  equal  parts,  such  that  each  shall  be  less 
than  0/;  there  will  be  at  least  one  point  of  division  m^ 
between  0  and  L  Let  P  denote  the  parallelopipedon, 
whose  base  is  ABCD^  and  altitude  Am ;  since  the  altitudes 
AE^  Am^  are  to  each  other  as  two  whole  numbers,  we 
have 

sol.  AG     :     P    ::     AE    :     Am. 
But  by  hypothesis,  we  have 

sol.  AG    :     sol.  AL    '.'.     AE    :     AO  ] 
therefore  (b.  ii.,  p.  4), 

sol.  AL     :     P    :  :     AO     :     Am. 

But  ^0  is  greater  than  Am  \  hence,  if  the  proportion  is 
correct,  the  solid  AL  must  be  gTeater  than  P.  On  the 
contrary,  however,  it  is  less  :  therefore,  AO  cannot  be 
greater  than  AL.  By  the  same  mode  of  reasoning,  it  may 
be  shown  that  the  fourth  term  cannot  be  less  than  AL  \ 
therefore,  it  is  equal  to  AL -.  hence,  rectangular  parallelo- 
pipedons having  equal  bases,  are  to  each  other  as  their 
altitudes. 


BOOK    VII. 


189 


PEOrOSITION    XII.     THEOREM. 

Two  itctangular  parallelopipedons^  liaving  equal  altitudes^  arc  it 
each  other  as  their  bases. 

Let  tlie  parallelopipedons  AG,  AK,  have  the  same  alti- 
tude AE]  then  will  they  be  to  each  other  as  their  bases 
AC,  AK 


For,  having  placed  the 
two  solids  by  the  side  of 
each  other,  as  the  figure 
represents,  prolong  the  plane 
NKLO  till  it  meets  the  plane 
DGGII  in  PQ ;  we  thus 
have  a  third  parallel  opipe- 
don  AQ,  which  may  be 
compared  with  each  of  the 
parallelopipedons  AG\,  AK. 
The  two  solids  AG,  AQ, 
having  the  same  base  ADHE 
are  to  each  other  as  their 
altitudes  AB,  A  0 :   in  like 


L 


H 


Y 


M 


k 


X 


4- 


N 


k 


'hvi 


o 


.p.  i 


B 


manner,  the  two  solids  AQ,  AK,  having  the  same  base 
A  OLE,  are  to  each  other  as  their  altitudes  AI),  AM. 
Hence,  we  have 

sol  AG    :     sol  AQ    ::     AB    :     AG; 
also,  sol  AQ    :     sol  AK    :  :     AD    :     AM. 

Multiplying  together  the  corresponding  terms  of  these  pro- 
portions, and  omitting,  in  the  result,  the  common  multi- 
plier sol  AQ;   we  shall  have 

sol  AG    :     sol  AK    :  :     ABxAD    :     AOxAK 

But  ABxAD  represents  the  area  of  the  base  ABCD;  and 
AOxAM  represents  the  area  of  the  base  AMNO;  hence, 
two  rectangular  parallelopipedons  having  equal  altitude^ 
are  to  each  other  as  their  bases. 


190 


GEOMETEY, 


rROPOSITIOX   XIII.     TILEOEEM. 


Any  Uco  rectangular  parallelopipedons  are  to  each  other  as  the 
2:>roducts  of  their  bases  hy  their  altitudes ;  that  is^  as  iJie 
products  of  their  three  dimensions. 


Having   i^laced  the  two 
solids    AG^    AZ^     so     that 
their    faces   have   the  com- 
mon  angle   BAE,   produce 
the     planes    necessary    for 
completing   the    third   par- 
allelopipedon    AK^     Avhich 
will  have  an  equal  altitude 
with    the    parallelopipedon 
AG.     By  the  last  proposi- 
tion, we  have 
sol  AG     :     solAK    :  : 
ABCD    :     AJIXO. 
But  the   two  parallelopipe- 


k 


Yr 


i 


n 


XI 


r rhG 


I 

by] 


N^ 


O 


Di ; 


... i 

p 


B 


dons  AK,  AZ,  having  the  same  base  NA^  are  to  each  other 
as  their  altitudes  AF,  AX;    hence,  Ave  have, 

sol  AK    :     so?.  AZ    :  :     AE    -.     AX. 
Multiplying  together  the  corresponding  terms  of  these  pro- 
portions, and  omitting  in  the  result  the  common  multiplier 
sol  AK  \    we  shall  have, 

sol  AG     :     solAZ    ::     ABCDxAE    :     AMXOxAX. 
Instead  of  the  bases  .li?^  and  .-iJ/.V6»,  put  ABxAD  and 
AOxAM^  and  we  shall  have, 

sol  AG  :  solAZ  ::  ABxADxAE  :  AOxAMxAX- 
hence,  any  two  rectangular  parallelopipedons  are  to  eaci 
other,  as  the  products  of  their  three  dimensions. 

Scholium  1.  The  magnitude  of  a  solid,  its  volume  oi 
extent,  is  called  its  solidity ;  and  this  word  is  exclusively 
emploved  to  designate  the  measure  of  a  solid :  thus,  we 
say  the  solidity  of  a  rectangular  parallelopipedon  is  equal 
to  the  product  of  its  base  by  its  altitude,  or  to  the  product 
of  its  three  dimensions. 


BOOK    YII.  191 

In  order  to  comprehend  the  nature  of  this  measurement, 
it  is  necessary  to  consider,  that  the  number  of  linear  units 
in  one  dimension  of  the  base  multiplied  by  the  number  of 
linear  units  in  the  other  dimension  of  the  base,  will  give 
the  number  of  superlicial  units  in  the  base  of  the  parallel- 
opipedon  (b.  IV.,  P.4,  s.)  For  each  unit  in  height,  there  are 
evidently,  as  many  solid  units  as  there  are  superficial  units 
in  the  base.  Therefore,  the  number  of  superlicial  units  in 
the  base  multiplied  by  the  number  of  linear  units  in  the 
altitude,  gives  the  number  of  solid  units  in  the  parallclo- 
pipedon. 

If  then,  we  assume  as  the  unit  of  measure,  the  cube 
whose  edge  is  equal  to  the  linear  unit,  the  solidity  Avill  be 
expressed  numerically,  by  the  number  of  times  which  the 
solid  contains  its  unit  of  measure. 

ScJioIiwji  2.  As  the  three  dimensions  of  tlie  cube  are 
equal,  if  the  edge  is  1,  the  solidity  is  1x1x1  =  1:  if 
the  edge  is  2,  the  solidity  is  2x2x2=8;  if  the  edge 
is  3,  the  solidity  is  3XoX3=27  ;  and  so  on.  Hence, 
if  the  edges  of  a  series  of  cubes  are  to  each  other  as  the 
numbers  1,  2,  3,  &c.,  the  cubes  themselves,  or  their  solidi- 
ties, are  as  the  numbers  1,  8,  27,  &c.  Ilence  it  is,  that 
in  arithmetic,  the  cube  of  a  number  is  the  name  given  to 
a  product  which  results  from  three  equal  factors. 

If  it  were  proposed  to  find  a  cube  double  of  a  given 
cube,  we  should  have,  unity  to  the  cube-root  of  2,  as  the 
edge  of  the  given  cube  to  the  edge  of  the  required  cube. 
Now,  by  a  geometrical  construction,  it  is  easy  to  find  the 
square  root  of  2  ;  but  the  cube-root  of  it  cannot  be  found, 
by  the  operations  of  elementary  geometry,  which  are  limit- 
ed to  the  employment  of  the  straight  line  and  circle. 

Owing  to  the  difficulty  of  the  solution,  the  problem 
of  the  duplication  of  the  cube  became  celebrated  among  the 
ancient  geometers,  as  well  as  that  of  the  trisection  of  an 
a??^fe  wdiich  is  a  problem  nearly  of  the  same  species.  The 
solutions  of  these  problems  have,  however,  long  since  been 
discovered  ;  and  though  less  simple  than  the  constructions 
of  elementary  geometry,  thej  are  not,  on  that  account,  less 
rigorous  or  less  satisfactory. 


192  GEOMETKY, 


PKOPOSITION   XIV.     THEOREM 

The  solidily  of  a  2^<^'/'ciJIeIo2:>i2yedon,  and  generally  of  any  prisrn^ 
IS  equal  to  the  jyroduct  of  its  base  hy  its  altitude. 

First.  Any  parallelopipedon  is  equivalent  to  a  rectau- 
galar  parallelopipedon.  having  an  equal  altitude  and  an 
equivalent  base  (p.  10).  But,  tlie  solidity  of  a  rectangular 
parallelopipedon  is  equal  to  its  base  multiplied  by  its 
height ;  hence,  the  solidity  of  any  parallelopipedon  is  equal 
to  the  product  of  its  base  by  its  altitude. 

Second.  Any  triangular  prism  is  half  a  parallelopipedon 
so  constructed  as  to  have  an  equal  altitude  and  a  double 
base  (p.  7).  But  the  solidity  of  the  parallelopipedon  is 
equal  to  its  base  multiplied  by  its  altitude ;  hence,  that  of 
the  triangular  prism  is  also  equal  to  the  product  of  its  base, 
Avhich  is  half  that  of  the  parallelopipedon,  multiplied  into 
its  altitude. 

Third.  Any  prism  may  be  divided  into  as  many  trian- 
gular prisms  of  the  same  altitude,  as  there  are  triangles 
formed  by  drawing  diagonals  from  a  common  vertex  iu 
the  polygon  which  constitutes  its  base.  But  the  solidity 
of  each  triangular  prism  is  equal  to  its  base  multiplied  by 
its  altitude ;  and  since  the  altitudes  are  equal,  it  follows 
that  the  sum  of  all  the  triangular  prisms  must  be  equal  to 
the  sum  of  all  the  triangles  which  constitute  their  bases, 
multiplied  by  the  common  altitude. 

Hence,  the  solidity  of  any  polygonal  prism,  is  equal  to 
the  product  of  its  base  by  its  altitude. 

Cor.  Since  any  two  prisms  are  to  each  other  as  the 
products  of  their  bases  and  altitudes,  if  the  altitudes  be 
equal,  they  will  be  to  each  other  as  their  bases  simply; 
hence,  two  j;/-^5??2s  of  the  same  altitude  are  to  each  other  as 
their  bases.  For  a  like  reason,  two  j^^'isnis  having  equivalent 
bases  are  to  each  other  as  Hieir  altitudes. 


LOOK    YII. 


193 


riiorosiTiox  xv.  theorem. 


Two    trianguhtr    '■pyramids^  having   equicahyit    hashes    and   cquaJ 
alttt/idcs,  are  eqiiicalcul^  or  equal  in  solid/'tf/. 

Let  S-AIl(\  S-uhr^  be  two  such  pyramids;  let  their 
equivalent  bnscs  AlUJ,  ahr^  be  situated  in  the  same  plane, 
and  let  .-1  7'  be  their  common  altitude  :  then  will  they  bf^ 
equivalent. 


For,  if  these  pyramids  arc  not  equivalent,  let  S-abc  bo 
the  smaller;  and  sui>pose  Aa  to  be  the  altitude  of  a  ])rism 
v.'hich,  having  ABC  for  its  base,  is  equal  to  their  differ- 
ence. 

Divide  the  nltitude  AT  into  equal  parts  Ax,  xjj,  yz,  ko^ 
each  less  than  Aa,  and  let  k  denote  one  of  tlujse  parts; 
through  the  points  of  division  j)ass  planes  parallel  to  the 
planes  of  the  bases ;  the  cori'esponding  sections  formed  by 
these  j)lanes  in  the  two  j^yramids  are  respectively  equiva^ 
lent,  namely,  JJFF  to  drf,  GUI  to  gin]  &c.  (i».  3,  a  2). 

This  being  done,  upon  the  triangles  ABC,  1)K1'\  GIII^ 
kc,  taken  as  bases,  construct  exterior  prisms  having  for 
edges  the  parts  AD,  DG,  GK,  &e.,  of  the  edge  SA  ;  in  lik© 
manner,  on  bases  def,  ghi,  him,  kc,  in  the  second  pyramid, 
construct  interior  prisms,  having  for  edges  the   corrcspond- 

13 


19-1 


GEOMETRY. 


ing  parts  of  Sa.  It  is  plain,  that  the  sum  of  all  the  exte- 
rior ])risms  of  the  pyramid  S-ABC  is  greater  than  this 
pyramid  ;  and  also,  that  the  sum  of  all  the  interior  prisms 
of  the  pyramid  S-ahc  is  less  than  this  pyramid.  Ilence, 
the  difference,  between  the  sum  of  all  the  exterior  prisms 
of  one  pyramid,  and  the  sum  of  all  the  interior  prisms  of 
the  other,  is  greater  than  the  difference  between  the  two 
pyramids  themselves. 

Kow,  beginning  with  the  bases,  the  second  exterior  prism 
EFD-G^  is  equivalent  to  the  first  interior  prism  efd-a, 
because  they  have  the  same  altitude  Z:,  and  their  bases 
EFD,  efd^  are  equivalent;  for  a  like  reason,  the  third  exte- 
rior prism  IIIG-K^  and  the  second  interior  prism  hig-d  are 
equivalent;  the  fourth  exterior  and  the  third  interior;  and 
so  on,  to  the  last  in  each  series.  Hence,  all  the  exterior 
prisms  of  the  pyramid  S-ABC,  excepting  the  first  prism 
BCA-J),  have  equivalent  corresponding  ones  in  the  interior 
prisms  of  the  pyramid  S-abc :  hence,  the  prism  BCA-D,  is 
the  diHerence  between  the  sum  of  all  the  exterior  prisms 
of  the  pyramid  S-ABC,  and  the  sum  of  the  interior  prisms 
of  the  jn'ramid  S-ahc.  But  the  difference  between  these 
two  sets  of  prisms  has  already  been  proved  to  be  greater 
than  that  between  the  two  pyramids  ;  which  latter  difference 
we  supposed  to  be  equal  to  the  prism  BCA-a  :    hence,  the 


BOOK    yil.  195 

priso)  BCA-D,  should  be  greater  than  the  prisiri  BCA-a. 
But  in  reality  it  is  less;  for  they  have  the  same  biise 
ABC,  and  the  altitude  Ax  of  the  first  is  less  than  the 
altitude  Act  of  the  second.  Ilence,  the  supposed  inequality 
between  the  two  pj'ramids  cannot  exist ;  therefore,  the  two 
pyramids  S'ABC\  S-abc,  having  equal  altitudes  and  equiva 
lent  bases,  are  themselves  equivalent. 

rKOPOSlTION    XVI.     THEOREM. 

Eue^'j  triangular  iirivn    may  he  dtcided    iido    three    equivalent 
triaufjnlar   'pyramids. 

Let  ABC-DEF  be   a  triangular  prism;    then  may  it  be 
divided  into  three  equivalent  triangular  pyramids.  , 

Cut  off  the  pyramid  /'-.:! i>'(7  -^  -pv 

from    the    prism,  by  the  plane  /"^X~  ~/\ 

FAC\    there    "will    remain    the  /      \/\  /  / 

solid   F-ACDE,  which   may  be  /  \  ^^F/        / 

considered    as   a    quadrangular  /  \<^l\        I 

pyramid,    Avhose    vertex    is    F,  /  y^  \'  \       / 

and  whose  base  is  the  parallel-  /       y^  /  \\    / 

o^^ram  A  CDF.     Draw  tlie  diao:-  ix  /        vj 

onal    CF;    and    pass   the  plane  \^^  /        / 

FCF^  which  will  cut  the  quad-  \^       /    / 

rangular     pyramid     into     two  ^^1/ 

triangular     ])yraniids     F-ACF^  ^ 

F-CJJF.  These  two  ti'iangnlar  pyramids  have  for  their 
common  altitude  the  perpendicular  let  fall  from  /•)  on  the 
plane  ACDE\  they  have  equal  bases;  for  the  triangles 
ACF,  CDF^  are  halves  of  the  same  parallelogram  ;  hence,  the 
two  pyramids  F-ACF,  F-CDFj  are  equivalent  (p.  15).  But 
the  pyramid  F^-CDE,  and  the  p3'ramid  F^-ABC^  have  equal 
bases  ABC^  DFF ;  they  have  also  the  same  altitude,  namely, 
the  distance  between  the  parallel  planes  ABC,  DBF;  hence, 
the  two  pyramids  are  equivalent.  Now,  the  pyramid  F-CDE, 
has  already  been  proved  equivalent  to  F-ACE\  hence,  the 
three  pyramids  F-ABC^  F-CDE^  F-ACE^  which  compose  the 
prism,  are  all  equivalent. 

Cor.  1     Every  triangular  pyramid  is   a  third  part   of   a 


196 


GEOMETKY. 


triangular    prism,  Avliicli    lias   an    equivalent   base    and    an 
equal  altitude. 

Cur.  2.    The   solidity  of  a  triangular   pyramid    is    equal 
to  a  third  paj-t  of  the  product  of  its  base  by  its  altitude. 


PKOrOSlTlON  XVII.     TIIEOKEM. 

The  soUdibj  of  every  pyramid  is  equal  to  a  tldrd  part  of  the 
irrodacl  of  its  base  by  its  altitude. 

Let  S- ABODE  be  a  pyramid  :  then  Avill  its  solidity  be 
equal  to  one-third  of  the  product  of  the  base  ABCIJE  by 
the  altitude  SO. 

Pass  the  jjlanes  SEB^  SEC,  througli 
the  vertex  S,  and  the  diagonals  EB,  EC; 
the  polygonal  p^-ramid  S-ABCDE  will 
then  be  divided  into  several  triangular 
j)yramids,  all  having  the  same  altitude 
SO.  But  each  of  these  pyramids  is  mea- 
.sured  by  the  jjroduct  of  its  base  ABE, 
BCE]  CUE,  by  a  third  part  of  its  alti- 
tude SO  {v.  U),  c.  2) ;  hence,  the  sum  of 
these  triangular  pyramids,  or  the  polyg- 
onal pyrandd  S-ABCDE  is  measured  by  tbe  sum  of  the 
triangles  ABE,  BCE,  CDE,  or  the  polygon  ABODE,  mul- 
tiplied  by  one-third  of  SO;  hence,  everv  p3'ramid  is  mea- 
sured by  a  third  part  of  the  product  of  its  base  by  its 
altitude. 

Cor.  1.    Every  pyramid    is    the    third    part  of    a    prism 
which  has  the  same  base  and  the  same  altitude. 

Cor.  2.    Two  p3'ramids   having   the  same  altitude  are  to 
each  other  as  their  bases. 

Cor.  3.    Two   pyramids  having   equivalent  bases   are    to 
each  other  as  their  altitudes. 

Cor.  4.    Pyramids  are  to  each  other  as  the  products  of 
their  bases  by  their  altitudes. 

Scholium.  The  solidity  of  any  polj'edral  body  may  be 
c<.Mnputed,  by  dividing   the   body  into  pyramids ;    and  tliis 


BOOK    YII 


197 


division  may  be  accomplished  in  various  ways.  One  of 
the  simplest  is  to  j)ass  all  the  planes  of  division  tliroufli 
the  vei-tex  of  the  same  polyedral  angle;  in  that  case,  there 
will  be  formed  as  many  pyramids  as  the  polyedron  has 
faces,  less  those  faces  Avhich  bound  the  polyedral  angle 
whence  the  planes  of  division  proceed. 


niOPOSITION  XVIII.     THEOREM. 

The  solidity  of  the  frustum  of  a  pyramid  is  equal  to  that  of 
three  ^:)?/ra??i/(/5  liaviug  for  their  common  altitude  tlie  alti- 
tude of  the  frustum.,  ami  for  bases  the  lower  base  of  the 
frustum^  tlie  upper  base,  and  a  mean  p)ro2^ortional  between 
the  two  bases. 

Let  ABCDE-e  be  the  frustum  of  a  pyramid  :  then  will 
its  solidity  be  equal  to  that  of  three  pyi-amids  having  the 
con:inion  altitude  of  the  frustum,  and  fur  bases  the  poly- 
gons ABCDI'J,  alicde^  and  a  mean  ])ro]*ortional  between 
them.  Let  T-FGIl  be  a  triangulnr  pyramid  having  the 
same  altitude,  and  an  equivalent  base  with  the  ])vramid 
jS-A/jCIJA\     These  two  ])yraniids  are  equivalent  (p.  17,  C.  o). 

Now,  if  we  regard  their 
bases  as  situated  in  the 
same  plane ;  the  plane  of 
the  section  abed,  will  form 
in  the  ti'iangular  pyramid  a 
section  /y//,  at  the  snme 
distance  above  the  common 
plane  of  the  bases ;  and, 
therefore,  the  section  f/h 
will  be  to  the  section  aUde,  as  the  base  FGII  is  to  the  base 
ABCIJE  (p.  8,  C.  1):  and  since  the  bases  are  equivalent, 
the  sections  will  also  be  equivalent.  Hence,  the  ])yramids 
S-abede^  T-fjh  will  be  equivalent  (p.  17,  C.  o).  If  these  be 
taken  from  the  entire  pyramids  S-AP>CJJI%  T-F(HJ,  the 
frustums  AUClJE-e,  F(i  H-h  which  remain,  will  be  ecpiiva- 
lent:  hence,  if  the  proi)()sition  is  ti'ue,  in  the  single  case  of 
the  fi'ustum  of  a  triangular  p3-ramid,  it  is  ti'Uo  :n  Ki\<iYy 
otlier. 


198 


GEOMETRY. 


Let  FGTI-h  be  the  frustum  of  a 
triangilar  ])yrainid.  Through  the 
three  points,  t\  y,  //,  pass  the  plane 
FqII\  it  cuts  oir  from  the  frustum 
the  triaiiguhir  pyramid  g-FGlI.  This 
pyramid  has  for  its  base  the  lower 
base  FGII  of  the  frustum  ;  its  alti- 
tude is  equal  to  that  of  the  frus- 
tum, because  the  vertex  g  lies  in  the 
plane  of  the  upper  base  fgh. 

This  pyi-amid  being  cut  off,  there  remains  the  quadran- 
gular pyramid  g-fhllF^  whose  vertex  is  g,  and  b^se  f/iIIF. 
Pass  the  plane  ^///  through  the  three  points  /  ^7,  //;  it 
divides  the  quadrangular  p3'ramid  into  two  triangular 
pyramids  g-fFlI,  g-fhlL  The  latter  hns  for  its  base  the 
Tipper  base  gfh  of  the  frustum  ;  and  for  its  altitude,  the 
altitude  of  the  frustum,  because  its  vertex  //  lies  in  the 
lower  base.  Thus  we  already  know  two  of  the  three  pyra- 
mids which  compose  the  fj'ustum. 

It  remains  to  examine  the  tliii-d  pyramid  g-F/TL  ISTow, 
if  gK  be  drawn  parallel  to  //%  and  if  we  conceive  a  new 
pyramid  K-JFII^  having  K  for  its  vertex  and  JFll  for  its 
base,  these  two  ])yrainids  have  the  same  base  lljF\  they 
also  have  the  same  altitude,  because  their  vertices  g  and 
K  lie  in  the  line  gK^  parallel  to  FJ^  and  consequently, 
parallel  to  the  ])lane  of  the  base:  hence,  these  pyramids 
are  equivalent  (i'.  17,  c.  8).  But  the  pyramid  K-j'Ftl  may 
be  regarded  as  having  FKIl  for  its  base,  and  its  vertex 
at/:  its  altitude  is  then  the  same  as  that  of  the  frustum. 
"We  are  now  to  show  that  the  base  FKll  is  a  mean  ])ro- 
portional  between  the  bases  FGll  and  fgh.  The  triangles 
FIIK^  fgh,  have  the  angle  F=f\    hence   (b.  iv.,  p.  24), 

FIIK    :    fgh     ::     FKxFII    :    fgXfh', 
but  because  of  the  parallels,  FK=fg, 

FIIK    .    fjh    :  :     FII    :    fh. 
We  have  also, 

FUG     :     FIIK    :  :     FG     :     FK  or  fg. 
But  the  similar  triangles  FGII,  fjh,  give 


BOOK  VII. 


199 


FG    :    fg     ::     FII    :    fh-, 
ueuce,  FGII    :     FUK    ::     FIIK    :    fgh; 

tliat  is,  the  base  FIIK  is  a  mean  proportional  between  the 
two  bases  FGII^  fgh.  Hence,  the  solidity  of  the  frustum 
of  a  triangular  pyramid  is  equal  to  that  of  three  pyramids 
whose  common  altitude  is  that  of  the  frustum,  and  whose 
bases  are  the  lower  base  of  the  frustum,  the  upper  base, 
and  a  mean  proportional  between  the  two  bases. 


PROPOSITION   XIX.     THEOREM. 

/Similar    triangular  prisms   are    to   each   other   as   the  cubes  of 
their  homologous  edges. 

Let  CBD-P,  chd-p^  be  two  similar  triangular  prisms,  and 
BC,  bcj  two  homologous  edges :  then  will  the  prism  CBD-P 
be  to  the  prism  chd-p,  as  BG    to  he  . 

For,  since  the  prisms  are  p 

similar,  the  homologous 
angles  ^  and  Z>  are  equal, 
and  the  flices  which  bound 
them  are  similar  (d.  16). 
Hence,  if  these  triedral  angles 
be  applied,  the  one  to  the 
other,  the  angles  chd  will 
coincide  with  CBD^  the  edge  ha  with  BA,  and  the  prism 
chd-p  will  take  the  position  Bcd-p.  From  A  draw  zL 7/ per- 
pendicular to  the  common  base  of  the  prisms :  then  will 
the  plane  BAH  be  perpendicular  to  the  plane  of  the  com- 
mon base  (b.  VI.,  p.  16).  Through  a,  in  the  plane  BAII^ 
draw  cdi  perpendicular  to  BII :  then  will  ah  also  be  per- 
pendicular to  the  base  BDC  (b.  vi.,  p.  17);  and  All,  ah  will 
be  the  altitudes  of  the  two  prisms. 

Since  the  bases  CBD^  chd,  are  similar,  we  have  (b.  iv.,  p.  25)^ 


hase   CBD 


hase  chd 


CB' 


ch\ 


Now,  because   of  the   similar   triangles  ABII,  aBh,   and  of 
the  similar  parallelograms  AC,  ac,  we  have 

All    :     ah     ::     AB    :     ah    ::    CB    :     ch; 

hence,  multiplying  together  the  corresponding  terms,  we  have 

hase  CBDxAII    :     hase  chd  X  ah     ::      CB^     :     d^. 


200 


GEOMETRY. 


But  the  solidity  of  a  prism  is  equal  to  tlie  base  multiplied 
by  tlie  altitude  (i\  14) ;    licncc, 

prii>m  BCD-P    :    ^)r^??i        hcd-p     :  :     JJC     :      he , 
or  as  the  cubes  of  anv  other  of  their  liomoloQ:ous  edsres. 

Cor,  Whatever  be  the  bases  of  similar  |)risms,  the 
prisms  are  to  each  other  as  the  cubes  of  their  homologous 
edges. 

For,  since  the  prisms  are  similar,  their  bases  are  simi- 
lar polygons  (d.  16) ;  and  these  similar  polygons  may  each 
be  divided  into  the  same  number  of  similar  triangles,  sim- 
ilarl}''  placed  (b.  iv.,  r.  26) ;  therefore,  each  prism  may  be 
divided  into  the  same  number  of  triang-ular  prisms,  having 
their  faces  similar  and  like  placed  ;  hence,  their  polyedral 
angles  are  equal  (b.  vi.,  i'.  21,  s.  2) ;  and  consequently,  the 
triangular  prisms  are  similar  (i).  16).  But  these  triangular 
prisms  are  to  each  other  as  the  cubes  of  their  homolo- 
gous edges,  and  being  like  parts  of  the  polygonal  prisms, 
their  sums,  that  is,  the  polygonal  prisms,  are  to  each  other 
as  the  cubes  of  their  homolo^'ous  edo-es. 


rEorosiTiox  xx.    tiieoeem. 

Tiuo  simikw  pyramids   are    to    each  other  as  the  cubes  of  their 
horaolorjous  edges. 

For,  since  the  pyramids  are  similar,  the  homologous 
polyedral  angles  at  the  vertices  are  equal  (d,  16).  Hence, 
the  polyedral  angles  at  the  vertices  may  be  made  to  coin- 
cide, or  the  tv»'0  pyramids  may  be  so  placed  as  to  hav^e 
the  polyedral  angle  S  common. 

In  that  position  the  bases  ABODE, 
abcde^  are  parallel  ;  fur,  the  homologous 
faces  being  similar,  the  angle  Sab  is  equal 
to  SAB,  and  ^Sbc  to  SBC]  hence,  the  plane 
ABC,  is  parallel  to  the  plane  obc  (b.  vi., 
p.  13).  This  being  proved,  let  SO  be 
drawn  from  the  vertex  S,  perpendicu- 
lar to  the  plane  ABQ  and  let  o,  be  the 
point  where  this  perpendicular  pierces  the 
plane  aba  :,  from    what   has   already  been 


BOOK 

VII. 

shown,  we  have  (p.  3), 

SO    :     So     ::     SA     : 

Sa     :  : 

AB    . 

ah 

and  consequently, 

ISO    :     iSo    : 

:     AB 

:     ah. 

201 


But  the  bases  ABCDE^  ahcde^  being  similar  figures,  \vf3 
have  (i3.  IV.,  P.  27), 

ABODE     :     ahcde     :  :     A'b'     :     ah'  ; 

multiplj  tlie  corresponding  terms  of  these  two  proportions,' 
there  results, 

ABODE  xlSO    :     ahcdt-xlSo     ::     IB''    :     «7A 

JSTow,  ABODE X^SO  measures  the  solidity  of  the  pyroniid 
S-ABOIJE,  and  ((bcdeX^So  measures  tluit  of  tlie  ju'ramid 
S-abcde  (i\  17) ;  hence,  two  similar  pyramids  are  to  each 
other  as  the  cubes  of  their  homologous  edges. 


GENERAL    SCirOIJU:\IS. 

1.  The  chief  propositions  of  this  Book  relating  to  the 
solidity  of  ])olyedi-ons,  may  be  expressed  in  algebraical 
terms,  and  so  recn])ituhited  in  the  briefest  manner  possible. 

2.  Let  B  represent  the  base  of  a  prism ;  II  its  altitude  ; 
then, 

solidity  of  prism^^Z?  X //. 

8.  Let i? represent  the  base  of  a  ^)v/?'(<?5i/c/;  U  its  altitude: 
then, 

sol  i d i ty  of  pj^ram  i  d  =  ^  X  J  77. 

*  4.  Let  //  represent  the  altitude  of  the  frustum  of  a  pyra- 
mid^ having  the  pfrallel  bases  A  ar.d  B\  y/ AxB  is  the 
mean  proportional  between  those  bases ;    then 

solidity  of  frustum=l//(J+i?+A/ZxrS.) 

5.  In  hne,  let  P  and  j)  represent  the  solidities  of  ticc  similar 
iwisms  or  pijramids ;  A  and  a^  two  homologous  edges : 
then. 

P    :    p    :  :     A      :     a . 


BOOK    VIII 


THE  THREE  ROUXD  BODIES, 


DEFIXITIOXS, 


1.  A  Cylixder  is  a  solid  which  may  be  generated  by 
the  revolution  of  a  rectangle  ABCD^  turning  about  the 
immovable  side  AB. 

In  this  movement,  the  sides  AD,  BC^ 
continuing  always  perpendicular  to  AB^ 
describe  the  equal  circles  VHP,  CGQ, 
-which  are  called  the  bases  of  the  cylindtr; 
the  side  CD^  describing,  at  the  same 
time,  the  convex  surface. 

The  immovable  line  AB  is  called 
the  axis  of  the  cylinder. 

Every  section  MSKL^  made  in  the 
cylinder,  by  a  plane,  at  right  angles  to  the  axis,  is  a  circle 
equal  to  cither  of  the  bases.  For,  whilst  the  rectangle 
ABCD  turns  about  AB,  the  line  /iT,  perpendicular  to  AB^ 
describes  a  circle,  equal  to  the  base,  and  this  circle  is 
nothing  else  than  the  section  made  by  a  plane,  perpendic- 
ular to  the  axis  at  the  point  I. 

Every  section  QPIJG^  made  by  a  plane  passing  through 
the  axis,  is  a  rectangle  double  the  generating  rectangle 
ABCD. 

2.  Similar  Cyxixders  are  those  whose  axes  are  pro- 
portional to  the  radii  of  their  bases  :  hence,  they  are  gen- 
erated by^  similar  rectangles  (b.  iv.,  d.  1). 


BOOK    YIII. 


203 


3.  If,  in  the  circle  ABODE,  wliich 
forms  the  base  of  a  cylinder,  a  polygon 
ABODE  be  inscribed,  and  a  right  prism, 
constructed  on  this  base,  and  equal 
in  altitude  to  the  cylinder  ;  then,  the 
prism  is  said  to  be  inscribed  in  tJie  cylin- 
der, and  the  cylinder  to  be  circurnscriled 
about  the  'prisrn. 

The  edges  AF,  BG,  Oil,  &c.,  of  the 
prism,  being  perpendicular  to  the  ])lane 
of  the  base,  are  contained  in  the  convex 
surface  of  the  cylinder ;  hence,  the 
prism    and   the   cylinder   touch    one    another 


>t 


V 


E\x 


:'D 


B 


along 


these 


edges. 


4.    In   like   manner. 


A 


\n^ 


MV 


O^ 


n 


B 


N 


if  ABOB  is  a 
polygon,  circumscribed  about  the  base 
of  a  cylinder,  a  right  prism  conptructed 
on  this  base,  and  equiil  in  altitude  to 
the  cylinder,  is  said  to  be  circumscribed 
about  the  cylinder,  and  the  cylinder  to  be 
inscribed  in  the  prism. 

Let  M,  N,  &c.,  be  the  points  of  con- 
tact in  tlie  sides  AB,  BO,  kc.  ;  and 
through  the  points  II,  N,  kc,  let  J/A", 
NY,  &c.,  be  drawn  perpendicular  to  the 
plane  of  the  base:  these  ])erpendiculars  will  then  lie  both 
in  the  surface  of  the  cylinder,  and  in  that  of  the  circum- 
scribed prism ;    hence,  they  will  be  their  lines  of  contact. 

5.  A  CoxE  is  a  solid  which  may  be  generated  by  the 
revolution  of  a  right-angled  triangle  SAB,  turning  about 
the  immovable  side  SA. 

In  this  movement,  the  side  AB  des- 
cribes a  circle  BDOE,  called  the  base  of 
the  cone  ;  the  hypothenuse  SB  describes 
the  convex  surface  of  the  cone. 

The  point  S  is  called  the  vertex  of 
the  cone,  SA  the  axis,  or  the  altitude,  and 
SB  the  slarU  hei(jht. 

Every    section    IIKFI,    made    by    a 


204: 


GEOMETEY. 


plane,  at  rigbt  angles  to  the  axis,  is 
a  circle.  Every  section  EDS^  made 
by  a  ])lane  passing  through  the  axis, 
is  an  isosceles  triangle,  double  the 
generating  triangle  SAB. 

G.  If,  from  the  cone  S-CDB,  the 
cone  S-FKH  be  cut  off  by  a  plane 
parallel  to  the  base,  the  remaining 
solid  CFIIB  is  called  a  truncated  cone, 
or  the  frustum  of  a  cone. 

The  frustum  may  be  generated  by  the  revolution  of  the 
trapezoid  ABIIG,  turning  about  the  side  AG.  The  im- 
movable line  AG  is  called  the  axis,  or  altitude  of  the  frustum^ 
the  circles  BUG,  IIFK,  are  its  bases,  and  BU  its  slant  lieight. 

7.  Similar  Coxes  are  those  whose  axes  are  propor- 
tional to  the  radii  of  their  bases  :  hence,  they  are 
generated    by    similar    ]ight-angled    triangles    (b.  iv.,  d.  1). 

8.  If,  in  the  circle  ABODE,  which 
forms  the  base  of  a  cone,  any  l><>ly- 
gon  ABODE  is  inscribed,  and  fi-om 
the  vei'tices  A,  B,  0,  J),  E,  lines  are 
drawn  to  »S',  the  vertex  of.  the  cone, 
these  lines  may  be  regarded  as  the 
edges  of  a  ])yramid  whose  base  is 
the  polygon  xiUODE  and  vertex  S. 
The  edges  of  this  pyramid  are  in  tlie 
convex    surface  of  the    cone,  and  the 

pyramid    is   said   to   be   inscrihed  in  the  cone.     The  cone  ia 
also  said  to  be  circumscribed  about  the  pyramid. 

9.  The  Si'iiERE  is  a  solid 
terminated  by  a  curved  sur- 
face, all  the  points  of  which 
are  equally  distant  from  a 
point  within,  called  the  centre. 

The  si)liere  may  be  gen- 
erated by  the  revolution  of 
a  semicircle  DAE,  about  its 
diameter  DE :  for,  the  surface 
descri])ed    in    this    movement. 


BOOK    Ylir.  205 

hy    the   semicircunifcrence   DAE,   will   have    all    its    points 
equal! J  Jitstaiit  Ironi  its  centre   C. 

10.  Whilst  the  semicircle  DAE,  rcv(jlving  round  its 
diameter  DE,  describes  the  sphere,  any  circular  seetor,  as 
DCE,  or  FCA,   describes  a  solid,  called  a  ^jjherical  aector. 

11.  The  radius  of  a  sphere  is  a  straight  line  drawn  from 
the  centi-e  to  any  j)oint  of  the  sui-face  ;  the  diameter  or  axis 
is  a  line  passing  through  the  centre,  and  terniinated,  on 
both  sides,  by  the  surface. 

All  the  radii  of  a  s})here  are  equal  ;  all  the  diameters 
are  equal,  and  each  is  double  the  radius. 

12.  It  will  be  shown  (p.  7,)  that  every  section  of  a 
sphere,  made  by  a  plane,  is  a  cii'cle:  tliis  gi'anted,  a  <jreat 
circle  is  a  section  whieh  passes  through  the  cenli'c;  a  6nLall 
circle,  is  one  which  does  not  pass  thi'ough  the  centi'e. 

13.  A  plf^ine  is  tangerd  to  a  sphere,  when  it  has  but  one 
point  in  common  with  the  surface. 

14.  A  zone  is  the  portion  of  the  surface  of  the  sphere 
included  between  two  parallel  circles,  which  form  its  buses. 
If  the  plane  of  one  of  these  circles  becomes  tangent  to  the 
t^phere,  the  zone  will  have  only  a  single  base. 

15.  A  spherical  srrjmejit  is  the  portion  of  the  solid  sphere, 
included  between  two  jnirallel  circles  whieh  Ibrm  its  bases. 
If  the  jJane  of  one  of  these  ciicles  becomes  tangent  to  the 
sphere,  the  segment  will  have  only  a  single  base. 

16.  The  altitude  of  a  zone,  or  of  a  ser/mcnt,  is  the  distance 
between  the  planes  of  the  two  parallel  circles,  which  form 
the  bases  of  the  zone  or  segment. 

17.  The  Cylinder,  the  Cone,  and  the  S})here,  are  the 
three  round  bodies  treated  of  in  the   Elements  of  Geumetrj. 


206 


GEOMETRY. 


rROPOSITION  I.     TnEOREil. 


The  convex   surface  of  a  cylinder  is  equal  to  the  circumference 
of  its  base  multiplied  by  its  altitude. 

Let  CA  be  the  radius  of  the  base  of  a  cylinder,  and 
11  its  altitude ;  denote  the  circumference  -whose  radius  is 
CA  by  circ.  CA :  then  Avill  the  convex  surface  of  the  cylin- 
der be  equal  to  circ.    CAxH. 

Inscribe  in  the  base  of  the 
cylinder  any  regular  polygon, 
BDEFGA,  and  construct  on 
this  polygon  a  right  prism 
having  its  altitude  equal  to 
II,  the  altitude  of  the  cylin- 
der:  this  prism  Avill  be  in- 
scribed in  the  cylinder.  The 
convex  suiface  of  the  prism 
is   equal    to    the   perimeter  of 

the  p<jlygon,  inultii)l)ed  by  the  altitude  H  (b.  vii.,  p.  1). 
Let  now  the  arcs  which  are  subtended  by  the  sides  of  the 
polygon  be  continually  bisected,  and  the  number  of  sides 
of  the  polygon  continually  doubled:  the  limit  of  the  perime- 
ter of  the  polygon  is  circ.  CA  (b.  5,  P.  12,  s.  2),  and  the  limit 
of  the  convex  suiface  of  the  prism  is  the  convex  surface 
of  the  cylinder.  But  the  convex  surface  of  the»prism  is 
alwavs  ecpial  to  the  pei'imeter  of  its  base  multij^lied  by 
//;  hence,  t/te  convex  surface  of  the  cylinder  is  equal  to  the 
circuiitfsrtiice  of  its  base  multij)lied  by  its  altitude. 


PKOPOSITION  II.     THEOREM. 

The  solidity  of  a  cylinder  is  equal  to  the  product  of  its  hose  by 

its  altitude. 


Let  CA  be  the  radius  of  the  base  of  the  cylinder,  and 
77  the  altitude.  Let  the  circle  whose  radius  is  CA  be 
denoted  by  area  CA  :  then  will  the  solidity  of  the  cylindei 
be  equal  to  area  CAxU. 


BOOK    YIII, 


207 


II 


For,  inscribe  in  the  base 
of  the  cylinder  any  regular 
polygon  BDEFGA^  and  con- 
struct on  this  polygon  a  right 
ju'ism  having  its  altitude  equal 
to  //,  tlie  altitude  of  the 
cylinder:  this  prism  will  be 
inscribed  in  the  cylinder.  The 
solidity  of  this  prism  will  be 
equal  to  the  area  of  the  poly- 
gon multiplied  by  the  altitude  //  (b.  vii.,  P.  14). 

Let  now  the  number  of  sides  of  the  polygon  be  con- 
tinually increased,  as  before  described ;  the  solidity  of 
each  new  prism  will  still  be  equal  to  its  base  multiplied 
by  its  altitude :  the  limit  of  the  polygon  is  the  area  CAj 
and  the  limit  of  the  prisms,  the  circumscribed  cylinder. 
But  the  solidity  of  each  new  prism  is  equal  to  the  base 
multiplied  by  the  altitude:  therefore,  the  solidity  of  the  cijlln- 
der  is  equal  to  the  -prodact  of  its  hase  hy  its  altitude. 

Cor.  1.  Cjdinders  of  equal  altitudes  are  to  each  other  as 
their  bases;  and  cylinders  of  equal  bases  are  to  each  other 
as  their  altitudes. 

Cor.  2.  Similar  cylinders  are  to  each  other  as  the  cubes 
of  their  altitudes,  or  as  the  cubes  of  the  radii  of  their 
bases.  For,  the  bases  are  as  the  squares  of  their  radii 
(b.  v.,  p.  13);  and  the  cylinders  being  similar,  the  radii  of 
their  bases  are  to  each  other  as  their  altitudes  (i).  2) ; 
hence,  the  bases  are  as  the  squares  of  the  altitudes;  there- 
fore, the  bases  multiplied  by  the  altitudes,  or  the  cylinders 
themselv^es,  are  as  the  cubes  of  the  altitudes. 

Scholium.   Let  R  denote  the  radius  of  a  C3dinder's  bai<e 
and  11  the  altitude ;    then  we  shall  have, 

surface   of  base=^Xi?^, 
convex  surface =2:^x7?  XU^ 
solidity ='^xBxII. 


20S 


GEOMETRY, 


TKOrOSITION    III.     TIIEOKEM. 

TJie  convex   surface  of  a  cone  is  equal  to  the  circumference  of 

its  base,   mullqdicd  bij  half  the  slant  lieijlit. 

Let  tlic  circle  A  BCD  be  the  base  of  a  cone.  S:  the 
vertex,  aSO  the  altitude,  and  >SA  the  slant  height:  then 
will  the  convex  surface  be  equal  to  circ.  OAx^^SA. 

For,  inscribe  in  the  base 
of  the  cone  an}'  regiilar  poly- 
gon A  BCD,  and  on  this  poly- 
gon as  a  base  conceive  a  right 
pyramid  to  be  constructed, 
havina:  ^S*  for  its  vertex  :  this 
pyramid  will  be  inscribed  in 
the  cone. 

From  S,  draw  SG  perpen- 
dicular   to    one  of  the    sides 

of  the  polygon.  The  convex  surface  of  the  inscribed  pyra- 
mid is  equal  to  the  perimeter  of  the  jjolygon  -which  forms 
its  base,  nmltijuied  by  half  the  slant  height  SG  (b.  VIL,  P.  4). 
Let  now  the  number  of  sides  of  the  inscribed  polygon  be 
continually  increased,  as  before  described:  the  limit  of 
the  pei-imeters  of  the  polygons  is  circ.  OA  ;  the  limit  of  the 
slant  height  of  the  ]\yramids  is  the  slant  height  of  the  cone, 
and  the  limit  of  their  sui'fnces,  is  the  convex  surface  of 
the  circumscribed  cone.  But  the  convex  surface  of  each 
new  pyramid  is  equal  to  the  perimeter  of  the  base  multi- 
plied by  half  the  slant  height  (b.  VIT.,  p.  4)  ;  hence,  the 
convex  sarface  of  the  cone  is  eqnal  to  the  circuwference  of  it3 
lose  rnulti piled  by  half  its  slant  height. 

Scholinm.   Let   L   denote   the    slant   height,    and   E  tlie 
radius  of  the  base  :    then, 

convex  surfixce=2^Xi?X^Z=-xi?X/v. 


BOOK    YIII. 


209 


rKOPOSlTION   IV.     TIIEOKEM. 


The  convex  surface  of  lite  frusi'/m  of  a  cone  is  equal  to  its 
slant  hei'jht,  muUijjlied  hij  half  the  sum  of  the  circumferences 
of  its  bases. 

Let  BIA-DE  be  a  frustum  of  a  cone :    tlien  will, 
convex  surfacc=.4/Jx J(c//-c.  OA-Vcirc.  CD.) 

For,  inscribe  in  the  bases  of 
tlie  frustum  two  regular  poly- 
gons of  tlie  same  number  of 
sides,  and  having  their  sides 
parallel,  each  to  each.  The  lines 
joining  the  vertices  of  the  corres- 
ponding angles  may  be  regarded 
as  the  edges  of  the  frustum  of  a 
right  pyramid  inscribed  in  the 
frustum  of  the  cone.  The  convex  surface  of  the  frustum 
of  the  pyramid  is  equal  to  half  the  sum  of  the  perimeters 
of  its  bases  multiplied  by  the  slant  height  fh  (b.  VIL, 
p.  4,  c.)  Let  the  number  of  sides  of  the  inscribed  polygons 
be  continually  increased  as  before  described :  the  limits 
of  the  perimeters  of  the  polygons  are  circ.  OA  and  circ. 
CD  ;  the  limit  of  the  slant  height  is  the  slant  height  of 
the  frustum,  and  the  limit  of  the  convex  surface,  the  con- 
vex surface  of  the  frustum ;  hence,  tJie  convex  surface  of  the 
frustum  of  a  cone  is  equal  to  its  slant  height  multiplied  by  half 
the  sum  of  tlie  circumferences  of  its  bases. 

Cor.  Through  Z,  the  middle  point  of  AD,  drcyw  IKL 
parallel  to  ^li?,  also  h\  Dd,  parallel  to  CO.  Then,  since  Al, 
ID,  are  equal,  Ai,  id,  are  also  equal  (b.  iv.,  p.  15,  c.  2): 
hence,  Kl  is  equal  to  \{OA-\-CD).  But  since  the  circum- 
ferences of  circles  are  to  each  other  as  their  radii  (b.  v., 
P.  13), 

circ.  Kl=\{circ.  OA+circ.  CD)\ 

therefore,  the  convex  surface  of  the  frustum  of  a  cone  is  equal 
to  its  slant  height  multiplied  by  tlie  circumference  of  a  section 
at  equal  distances  from  the  two  bases. 

14 


210 


GEOMETRY. 


Scholium  1.  If  from  the  mid- 
dle point  I  and  the  two  extrem- 
ities A  and  D^  of  a  line  AD^ 
lying  Avliolly  on  one  side  of  the 
Ihie  06'  the  perpendiculars  DO^ 
IK,  and  A  0,  be  drawn,  and  then 
the  line  AD  be  revolved  around 
OC,  we  shall  have 

surf  described  by  AI)=ADxl{circ.  OA+circ.  CD); 
that  is,  =ABxcirc.  KL* 

For,  it  is  evident  that  the  surface  described  by  AD  \?>  that 
of  the  frustum  of  a  cone,  having  OA  and  CI)  for  the 
radii  of  its  bases. 

/ScJioUum  2.  The  measure  found  above  applies  equally 
to  the  case  when  the  point  I)  foils  at  C\  and  the  surface 
becomes  that  of  a  cone ;  and  to  the  case  in  which  AJJ 
becomes  parallel  to  OC,  and  the  surface  becomes  that  of  a 
cylinder.  In  the  first  case,  CD  is  nothing :  in  the  second, 
it  is  equal  to  OA. 


PKOrOSITION  V.    TIIEOEEM. 

TJie  solidity  of  a  cone  is  equal  to  its  hase  multiplied  hy  a  third 
of  its  altitude. 

Let  SO  be  the  altitude  of  a  cone,  OA  the  radius  of  its 
base,  and  let  the  area  of  the  base  be  designated  by  area 
OA ;    then   will, 

solidity = area  OA  X  ^SO. 

Inscribe  in  the  base  of  the 
cone  any  regular  polygon  ^7?Zy'^/'^, 
and  join  the  vertices  A^  B,  C,&lc., 
with  the  vertex  /S  of  the  cone: 
then  will  there  be  inscribed  in 
the  cone  a  right  pyramid  having 
the  same  vertex  as  the  cone,  and 
having  for  its  base  the  polygon 
ABDEF.  The  solidity  of\his 
pyramid  will  be  equal  to  its  base  multiplied  by  one -third 
of  its  altitude  (b.  vii.,  P.  17). 


BOOK    YIII.  211 

Let  the  arcs  be  bisected  and  the  number  of  sides 
of  the  polygon  be  continually  increased:  the  limit  of  the 
polygons  will  be  the  area  OA^  and  the  limit  of  the  pyra- 
mids will  be  the  cone  Avhose  vertex  is  S:  hence,  ilie  solid 
ity  of  the  cone  is  equal  to  its  base  multiplied  hij  a  third  of  its 
altitude. 

Cor.  1.  A  cone  is  the  third  of  a  cylinder  havingr  the 
same  base  and  the  same  altitude  ;    whence  it  follows, 

1.  That  cones  of  equal  altitudes  are  to  each  other  as 
their  bases; 

2.  That  cones  of  equal  bases  are  to  each  other  as  their 
altitudes ; 

3.  That  similar  cones  are  as  the  cubes  of  the  diameters 
of  their  bases,  or  as  the  cubes  of  their  altitudes. 

Gjv.  2.  The  solidity  of  a  cone  is  equivalent  to  the 
solidity  of  a  pyramid  having  an  equivalent  base  and  the 
same  altitude. 

Sdiolium..  Let  R  be  the  radius  of  a  cone's  base,  U  its 
altitude ;    then, 

solidity = Jr  x  7? "  X  //. 

TROrOSITION   VI.     TIIEORE^L 

The  solidity  of  the  frustum  of  a  cone  is  equivalent  to  the  sum 
of  the  solidities  of  three  cones  icliose  common  altitude  is  the 
altilude  of  the  frustmn,  and  ichose  bases  are^  the  lower  base 
of  the  frustum,  the  vjjper  base  of  the  frustum,  and  a  mean 
2)r^portional  between  them. 

Let   AEB-CD    be    the   frustum   of  a   cone,  and  OP  its 
altitude ;    then    will    its    solidity  be 
equivalent  to 

J^X  OPx{Olj'+Fa'+OBxPC). 

For,  inscribe  in  the  lower  and 
upper  bases  two  regular  polygons 
having  the  same  number  of  sides, 
and  having  their  sides  parallel, 
each  to  each.  Join  the  vertices  of 
the  corresponding  angles,  and  there 


212 


GEOMETRY. 


will  tlien  be  inscribed  in  tlie 
frustum  of  the  cone,  the  frus- 
tum of  a  regular  pyramid.  The 
solidity  of  the  frustum  of  this 
pyramid  ^vill  be  equivalent  to 
three  p^-ramids  having  the  com- 
mon altitude  of  the  frustum,  and 
for  bases,  the  lower  base  of  the 
frustum,  the  upper  base  of  the 
frustum,  and  a  mean  proportional 
between  them  (b.  vii.,  p.  18). 

Let  the  number  of  sides  of  the  inscribed  polygons  be 
continually  doubled  by  the  methods  before  described :  the 
limits  of  the  polygons  will  be,  area  OB  and  area  PC\  and 
the  limit  of  the  frustums  of  the  p3-ramids  will  be  the  frustum 
of  the  cone:  the  expression  for  the  solidity  will  then  become: 

iOPxOBxPCXrf. 
hence,  the  solidity  of  the  frustum  of  the  cone  is  equivalent  to 
ItX  OPx{OB'+PC'+OBxPa) 


of  the  first  pyramid, 
of  the  second 
of  the  third 


PROrOSITION  VII.     THEOREM. 
Every  section  of  a  spliere^  made  hj  a  j'^^ane,  is  a  circle. 

Let  AMB  be    any    section    made    by    a    plane,    in    the 
sphere  whose  centre  is   C:    then  Avill  it  be  a  circle. 

For,  from  the  point  C,  draw 
CO  perpendicular  to  the  plane 
AMB]  and  different  lines  CJf, 
C2f,  to  different  points  of  the 
curve  AJIB,  which  terminates 
the  section. 

The    oblique    lines    CM,    CM, 
CA,  are  equal,  being  radii  of  the 
sphere  ;    hence,    they   pierce    the 
plane  AMB    at   equal  distances  from  the  perpendicular  CO 
(b.  yi.,  p.  5,  c.) ;    therefore,  all   the   lines  OJ/,   OM,   OB,  are 


BOOK    YIII.  213 

equal;  consequently,  the    section  AMB  is    a   circle,  Avliose 
centre  is  0. 

Cor.  1.  If  the  section  pass  through  the  centre  of  the 
sphere,  its  radius  will  be  the  radius  of  the  sphere ;  hence 
all  great  circles  are  equal. 

Cor.  2.  Two  great  circles  always  bisect  each  other;  for 
their  common  intersection,  passing  through  the  centre,  is  a 
diameter. 

Cor.  3.  Every  great  circle  divides  the  sphere  and  its 
surflice  into  two  equal  parts :  for,  if  the  two  parts  were 
separated  and  afterwards  placed  on  the  common  base, 
with  their  convexities  turned  the  same  way,  the  two  sur- 
faces would  exactly  coincide,  no  point  of  the  one  being 
nearer  the  centre  than  any  point  of  the  other. 

Cor.  4.  The  centre  of  a  small  circle,  and  that  of  the 
sphere,  are  in  the  same  straight  line,  perpendicular  to  the 
plane  of  the  small  circle. 

Cor.  5.  The  radius  of  any  small  circle  is  less  than  the 
radius  of  the  sphere;  and  the  further  its  centre  is  remov- 
ed from  the  centre  of  the  sphere,  the  less  is  its  radius: 
for,  the  greater  CO  is,  the  less  is  the  chord  AB^  the  diam- 
eter of  the  small  circle  AMB. 

Cor.  6.  An  arc  of  a  great  circle  may  alwavs  be  made 
to  pass  through  any  two  given  points  of  the  surface  of  the 
sphere:  for,  the  two  given  points,  and  the  centre  of  the 
sphere  make  three  points,  which  determine  the  {)ositi()n  of 
a  plane.  But  if  the  two  given  points  were  at  the  extremi- 
ties of  a  diameter,  these  two  points  and  the  centre  w<iu]d 
then  lie  in  one  straight  line,  and  an  induite  number  of 
great  circles  might  be  made  to  pass  through  the  two  given 
points. 

Cor.  7.  The  distance  beiweeri  any  two  j^oints  on  the  surftre  of 
a  sphere  is  less  iclien  measured  on  the  arc  of  a  great  circle 
than  ivlien  measured  on  the  arc  of  a  small  circle. 

For,  let  A  and  B  be  any  two  points  on  the  surface  of 
a  sphere,  let  ADB  be  the  arc  of  a  great  circle,  and  A  ME 


2U  GEOMETRY. 

the  arc  of  a  small  circle  passing  tlirough  them,  and  AB 
the  common  chord.  Then,  since  the  radius  CA  is  greater 
than  the  radius  OA,  the  arc  ADB  is  less  than  the  arc 
AJIB  (b.  v.,  p.  17). 


PEOrOSITIOX  Ylll.     THEOREM. 

Eueri/  2)lane  perj^endicular  to  a  radius  at  its  extremity  is  tan- 
gent to  the  sjjliere. 

Let  FAG  be  a  plane  perpendicular  to  the  radius 
OA^  at  its  extremity  A  :  then  will  it  be  tangent  to  the 
sphere. 

For,  assuming  any  other  point 
i/  in  this  plane,  draw  OA,  OM: 
then  the  ansrle  0AM  is  a  risht 
angle,  and  hence,  the  distance  OM 
is  greater  than  OA  :  therefore,  the 
point  M  lies  without  the  sphere; 
hence,  the  plane  FAG,  can  have 
no  point  but  A  common  to  it 
and  the  surfoce  of  the  sphere; 
consequently,  it  is  a  tangent  plane  (d.  13). 

Scholium.  In  the  same  way  it  may  be  shown,  that  two 
spheres  are  tangent  the  one  to  the  other,  when  the  dis- 
tance between  their  centres  is  equal  to  the  sum  or  the 
difference  of  their  radii  ;  in  which  case,  the  centres  and 
the  point  of  contact  lie  in  the  same  straight  line. 

TEOFOSITION  IX.      LEMMA. 

If  a  regular  semi-jyolggon  he  revolved  about  a  line  passing 
Virough  the  centre  and  the  vertices  of  two  opposite  angles, 
Vie  surface  described  by  its  perimeter  loill  be  equal  to  the 
axis  multiplied  by  the  circumference  of  the  inscribed  circle. 

Let  the  regular  semi-polygon  ABCDEF,  be  revolved 
about  the  line  A  F  as  an  axis :  then  will  the  surface  des- 
cribed by  its  perimeter  be  equal  to  AF  multi2)lied  by  the 
circumference  of   the  inscribed  circle. 


BOOK    VIII. 


215 


t 

^ 

F 
H 

M 

I 

n/1 

■n 

n. 

K  \ 

^ 

0 

C 

r 

\ 

^ 

Q 

A 

For,  from  E  and  i),  the  extremities 
of  one  of  the  equal  sides,  let  fall  the 
perpendiculars  EH^  DI^  on  the  axis  AE] 
and  from  the  centre  0,  draw  OX  per- 
pendicular to  the  side  DE :  OxY  will  be 
the  radius  of  the  inscribed  circle  (b.  v., 
p.  2).  Now,  the  surface  described  in  the 
revolution,  by  any  one  side  of  the  reg- 
ular polygon,  as  BE,  has  been  shown 
to  be  equal  to  DE  X  circ.  NM  (p.  4,  s.  1). 
But  since  the  triangles  EDK^  ONM^  are 
similar  (b.  iy.,  p.  21), 

EB  :  EK  or  ///  :  :  ON  :  NM  :  :  circ.  ON  :  circ.  NM] 
hence,  ED X circ.  NM= III X circ.  ON; 

and  since  the  like  may  be  shown  for  each  of  the  other 
sides,  it  is  plain  that  the  surface  described  by  the  entire 
perimeter  is  equal  to 

{EM+III+IP+PQ+QA)Xcirc.  ON=AEXcirc.  ON, 

Cor.  The  surface  described  by  any  portion  of  the  peri- 
meter, as  EDO,  is  equal  to  the  distance  between  the  two 
perpendiculars  let  fall  from  its  extremities  on  the  axis, 
multiplied  by  the  circumference  of  the  inscribed  circle. 

For,  the  surface  described  by  DE  is  equal  to  III  X  circ. 
ON,  and  the  surface  described  hy  DO  is  equal  to  IPxcirc 
ON:  hence,  the  surfoce  described  by  ED+DC,  is  equal  to 
{III+IP)Xcirc.  ON,  or  equal  to  IlPXcirc.  ON. 


PROPOSITION  X.     TIIEOEEM. 

The  surface  of  a  sphere  is  equal  to  the  product  of  its  diameter 
hy  the  circumference  of  a  great  circle. 


Let  ABODE  be  a  semicircle.  Inscribe  in  it  a  regu- 
lar semi-polygon,  and  from  the  centre  0  draw  OF  perpen- 
dicular to  one  of  the  sides. 

Let  the  semicircle  and  the  semi-polygon  be  revolved 
about  the  common  axis  AE :  the  semicircumference  ^^C7)^ 
will  describe  the  surface  of  a   sphere  (d.  9) ;    and  the  peri- 


216 


GEOMETRY 


meter  of  the  semi-polygon  will  describe 
a  surface  which  has  for  its  measure 
AE  X  circ.  OF  (p.  9),  and  this  will  be 
true  whatever  be  the  number  of  sides  of 
the  semi -polygon. 

If  now,  the  arcs  be  continually  bisected, 
the  limit  of  the  perimeters  of  the  semi- 
polj^gons  will  be  the  semicii'cumference 
ABODE \  the  limit  of  the  area  described 
by  the  perimeter  will  be  surface  of  the 
sphere,  and  the  limit  of  the  perpendicular  OF  Avill  be  the 
radius  OE:  hence,  the  surface  of  the  sphere  is  equal  to 
AEXcirc.  OE. 

Cor  1.  Since  the  area  of  a  great  circle  is  equal  to  the 
product  of  its  circumference  by  half  the  radius,  or  one- 
fourth  of  the  diameter  (u.  v.,  r.  15),  it  follows  that  tlie  sur- 
face of  a  siiihere  is  equal  to  fnir  of  its  great  circles :  that  is, 
equal  to  4rx  OA    (b.  v.,  p,  16). 

Cor.  2.  The  surface  of  a  zone  is  equal  to  its  altitude  mul- 
tiplied hy  tlie  circumftrence  of  a  great  circle. 

For,  the  surface  described  by  any  por- 
tion of  the  perimeter  of  the  inscribed 
polygon,  as  BC-rCB,  is  equal  to  EHx 
circ.  OF  (p.  9,  c.) :  and  Avhen  we  pass  to 
the  limit,  we  have  the  surface  of  the  zone 
equal  to  EHx  circ.  OA. 

Cor.  3.  When  the  zone  has  but  one 
base,  as  the  zone  described  by  the  arc 
ABCB^  its  surface  will  still  be  equal  to 
the  altitude  AE  multiplied  by  the  circum- 
ference of  a  great  circle. 

Cor.  4.  Two  zones,  taken  in  the  same  sphere  or  in 
equal  spheres,  are  to  each  other  as  their  altitudes;  and 
any  zone  is  to  the  surface  of  the  sphere  as  the  altitude  of 
the  zone  is  to  the  diameter  of  the  sphere. 


BOOK  YIII. 


217 


PROPOSITION  XI.     LEMMA. 

If  a  triangle  and  a  rectangle,  having  the  same  base  and  the 
same  altitude^  turn  togetlier  about  the  comvion  base,  the  solid 
generated  by  the  triangle  is  a  third  of  the  cylinder  generated 
hy  ike  rectangle. 

Let  BAC  be  a  triangle,  BFEC  a  rectangle,  having  the 
common  base   BC,   about   which   thej  are   to   be   revolved. 

On  the  axis,  let  fall  the  per- 
pendicular AD  :  then,  the  cone 
generated  by  the  triangle  BAD  is 
a  third  part  of  the  cylinder  gen- 
erated by  the  rectangle  BFAD  (p. 
v.,  c.  1) :  also,  the  cone  generated 
by   the   triangle   DAG  is   a   third 

part  of  the  cylinder  generated  by  the  rectangle  DA  EC: 
hence,  the  sum  of  the  two  cones,  or  the  solid  generated 
by  BACj  is  a  third  part  of  the  sum  of  the  cylinders  gen- 
erated by  the  two  rectangles,  or  a  third  part  of  the  cylinder 
generated  by  the  rectangle  BFEC. 

If  the  perpendicular  AD  falls 
without  the  triangle  ;  the  solid 
generated  by  CBA  is,  in  that 
case,  the  difference  of  the  two 
cones  generated  by  BAD  and 
CAD;  but  at  the  same  time,  the  cylinder  generated  by 
BFFC,  is  the  difference  of  the  two  cylinders  generated 
by  BFAD  and  CEAD.  Hence,  the  solid,  generated  b}^  the 
revolution  of  the  triangle,  is  still  a  third  part  of  the  cylin- 
der generated  by  the  revolution  of  the  rectangle  having 
the  same  base  and  the  same  altitude. 

Scholium.  The  circle  of  which  AD  is  the  radius,  has 
for  its  measure  'j^xAD  ;  hence,  it  xAD'xBC  measures  the 
cylinder  generated  by  BFEC,  and  ^^xAD'kBC  measures 
the  solid  generated  by  the  triangle  BAC. 


218  GEOMETRY. 

PKOPOSITION    XII.      LEilMA. 

If  a  triangle  he  revolved  about  any  line  draiini  tlirovgh  its  ver- 
tex ill  the  same  p/rt/?e,  the  solid  generated  will  have  for  its 
measure,  the  area  of  tJte  triangle  multiplied  by  two  thirds  of 
the  circumference  traced  by  the  middle  i^oint  of  the  base. 

Let  CAB  be  a  triangle,  /  the  middle  point  of  the  :a3e, 
and   CD   the   line   about  which  it  is  to  be  revolved:    then 
will  the  solid  generated  be  measured  by 
area   CAB  X  |  circ.  IK. 

Prolong  the  base  AB  till  it 
meets  the  axis  CD  hi  D  ]  from 
the  points  A  and  B,  draw  AJf, 
BX,  perpendicular  to  the  axis, 
and  draw  CP  perpendicular  to 
DA  produced. 

The  scholium  to  the  last  proposition  gives  the  following 
measures  : 

solid  generated  by  CAD=i-rXAll'x  CD, 
solid  generated  by   CBD=irrXBy'X  CD  : 

hence,  the  difference  of  these  solids,  which  is  the  solid 
generated  b}^  the  triangle   CAB,  has  for  its  measure 

irrX{All--BX'')xCD. 

To  this  expression  another  form  may  be  given.  From  ij 
the  middle  point  of  AB,  draw  IK  perpendicular  to  CD; 
and  through  B,  draw  BO  parallel  to  CD.  We  shall  then 
have  (b.  iv..  p.  7,  s.), 

AJI+BX=2IK,  and  AJiI-BX=AO', 

hence,  {AM-{-BX)x{AJI-BX)=A~M^'-BX'=2IKxA0 : 

hence,  the  measure  of  the  solid  is  also  equal  to 

irrxIKxAOxCD. 

But  CP  being  perpendicular  to  AB  produced,  the  triangles 
AOB  and   CPD  are  similar;    hence, 

AO     :     CP    ::     AB    :     CD. 
and,  AOxCD=CPxAB. 


BOOK    YIII, 


219 


But  CPxAB  is  double  the  area  of  tlie  triangle  CAB\ 
tlierefore, 

A0xCD^2CABi 

hence,  the  solid  generated  by  the  triangle  CAB  is  measured  by 

^<i^xCABxIK=CABx^^XlK\ 

and  since  2'irxIK=circ.  IK,  we  have, 

soYid=  CAB  Xlcirc.  IK 

Cor.  If  the  triangle  is 
isosceles,  the  perpendicular 
CP  will  pass  through  I, 
the  middle  point  of  the 
base ;    and  we  shall  have 

CAB=ABxl-CL 
Substituting    this    value  of 
CAB  in  the  measure  of  the  solid  before  found,  viz. : 
solid=  CAB  X  ^  -ff  X  IK,  gi  ves, 
sol  id = f  cr  X  ^1 B  X  IK  X  CI 
But  the  triangles  AOB^  CKI,  are  similar  (b.  iv.,  p.  21); 
hence,  AB    :     BO  or  MX    :  :     CI    :     IK, 

which  gives,  AB  x  IK=  MN x  CL 

Substituting  for  ABxIK,  we  have, 

solid-jT67'xJAY: 

that  is,  the  solid  generated  by  the  revolution  of  an  isos- 
celes triangle  about  any  line  drawn  through  its  vertex,  is 
measured  h/j  tico-lltirds  of  nt  into  the  square  of  /he  jyerpendicular 
let  fall  on  the  base,  into  the  distance  between  tJie  two  periyendic- 
ulars  let  fall  from  the  extremities  of  the  base  on  the  axis. 

Scholium.  The  demonstration  appears  to  involve  the  sup- 
position that  AB  prolonged  will  meet  the  axis:  but  the 
results  are  equally  true  if  AB  is  ])arallel  to  the  axis. 

Thus,    the    cylinder   generated   by       p  A  B 

MKBA  is  measured  b}^  -jt  x  AiWxJ^fy: 
the  cone  generated  by  CAM  is  mea- 
sured by  ^txAM'x  CM;  and  the 
cone  generated  by  CBN  is  measured 
by  irrxAH'xCX 


220 


GEOMETRY. 


P            A             B 

/^ 

-^ 

C            M            N 

Add  the  first  two  solids,  and  from  the  sum 
subtract  the  third :  we  shall  then  have 

solid  bj  CAB=^xATfx{}IN^ 
-cX.4l7'x(iJAY  + 
and  since  \}IX -\-\CM=\CN^  we  have 

solid  by  ai-s=crx.nz'xfj/iv: 

But  AM=CP  and  MX=AB-,   hence, 

solid  by  611J5=.4i?xCPxfcxaP=C45xfaVc.  CP. 
But  the  circumference  traced  by  P  is  equal  to  the  circum- 
ference traced  by  the  middle  i)oint  of  the  base :   hence,  the 
result  auTces  Avith  the  g-eneral  enunciation. 


rEOPOSITIOX    XIII.     LEilMA. 

Jf  a  regular  semi-pohjgon  he  revolved  about  a  line  passing 
through  its  centre  and  the  vertices  of  two  opposite  angles^  the 
solid  generated  ivill  he  measured  hij  tv:o-thirds  the  area  of 
the  inscrihtd  circle  rauliiplied  hy  tlte  axis. 

Let  GDBF  be  a  regular  semi-polygon  and  01  the  radius 
of  the  inscribed  circle :  then,  if  this  semi-polygon  be  re- 
volved about  GF^  the  solid  generated  will  have  for  its 
measure, 

iarea  OIxGF. 

For,  since  the  polygon  is  regular, 
the  triangles,  OF  A,  OAB,  OBC,  kc,  are 
isosceles  and  equal ;  then,  all  the  per- 
pendiculars let  fall  from  0  on  their 
bases,  will  be  equal  to  Oi,  the  radius 
of  the  inscribed  circle. 

Now,  we  have  the  following  mea- 
sures for  the  solids  generated  by  these 
triangles  (p.  12,  c.) :    viz., 

OFA  is  measured  by  I'jrxOI'xFM, 
OAB  "  "  "    i'^xOl'xJIX, 

OBC  "  "  ''    i^xClrxOX  &c.; 


BOOK    YIII. 


221 


hence,    the   entire   solid   generated   by  the  semi-poljgon   is 
measured  by 

f  ^X  01\fM+MN+N0^  OQ+QR+BG)  : 

that  is,  by  ^'rrXOI^GF. 

But,  'X  X  Ol'=area  01  (b.  v.,  p.  16) : 


hence. 


solidity  =  I  area   OIxGF. 


TEOPOSITION  XIV.     THEOREM. 

The  solidity  of  a  sphere  is  equal  to  its  surface  multiplied  hy  a 
third  of  its  radius. 

Let  0  be  the  centre  of  a  sphere  and  OA  its  radius : 
then  its  solidity  is  equal  to  its  surface  into  one-third  of 
OA. 

For,  inscribe  in  the  semi-circle 
ABODE  a  reguLar  semi-polj^gon,  hav- 
ing any  number  of  sides,  and  let  01 
be  the  radius  of  the  circle  inscribed  in 
the  polygon. 

If  the  semicircle  and  semi-poh^gon 
be  revolved  about  FA^  the  semicircle 
will  generate  a  sphere,  and  the  semi- 
polygon  a  solid  Avhich  has  for  its  mea- 
sure f -r  of  X  FA  (p.  13) ;  and  this  is 
true  whatever  be  the  number  of  sides 
of  the  semi-polvgoii.  But  if  the  number  of  sides  of  the 
polygon  be  continually  doubled,  the  limit  of  the  solids 
generated  by  the  poh'gons  will  be  the  sphere;  and  when 
we  T>ass  to  the  limit  the  expression  for  the  solidity  will 
become  ^'^XOA'XFA,  or  by  substituting  2  0A  for  .£".1,  it 
becomes  J-rx  O/FX  0^1,  which  is  also  equal  to  4:'tX0A'X 

o 

i  OA.  But  4-^x0^'  is  equal  to  the  surface  of  the  sphere 
(p.  X.,  c.  1) :  hence,  the  solidity  of  a  sphere  is  equal  to  it3 
Burface  multiplied  by  a  third  of  its  radius. 

Scholium  1.  The  solidify  of  every  spherical  sector  is  equal 
to  the  zone  which  forms  its  hase^  multip)lied  hy  a  third  of  tJie 
radius. 


222 


GEOMETRY, 


For,  the  solid  described  by  any 
portion  of  the  regidar  polygon,  as  the 
isosceles  triangle  OAB^  is  measured  by 
I'Ol'XAF  (p.  12,  c);  and  ^vhen  we 
pass  to  the  limit  which  is  the  spherical 
sector,  the  expression  for  this   measure 

o 

becomes  lr:xAO~XAF^  which  is  equal 

to  "IrrxAOxAFx^AO.     But  2crX.40 

is   the   circumference  of   a   great   circle 

of  the  sphere  (b.  v.,  p,  16),  which  being 

multiplied    by    AF    gives    the    surface 

of  the   zone   which   forms   the   base    of   the   sector    (p.  x, 

c.  2) ;    and  the  proof  is  equally  applicable  to  the   spherical 

sector    described    by  the   circular   sector    BOG:   hence,   the 

solidity  of  the  spherical  sector  is  equal  to  the  zone  ichich  forms 

its  base,  muUijjlied  hy  a  third  of  the  radius. 

Scholiimi  2.  Since  the  surfoce  of  a  sphere  whose  radius 
is  ii,  is  expressed  by  -^rrxR"  (p.  x.,  c.  1),  it  follows  that 
the  surfaces  of  spheres  are  to  each  other  as  the  squares  of 
their  radii ;  and  since  their  solidities  are  as  their  surfaces 
multiplied  hj  their  radii,  it  follows  that  tlte  solidities  of  spTteres 
are  to  each  other  as  tJte  cubes  of  their  radii,  or  as  the  cubes  of 
their  diameters. 

Scholium  3.    Let  R  be   the   radius  of  a   sphere ;    its   sur- 

o 

fiice   will    be    expressed    by    4^x7?^,    and    its    solidity   by 
4-Xi?~Xi7?,  or  J-rxytl     If  the  diameter  be  denoted  hj  D, 

3  3 

we  shall  have  R=\D,  and  R  =\D  :   hence,  the  solidity  of 
the  sphere  may  be  expressed  by 

4^XiZ>'-icxZ>l 


PKOPOSITION  XV.     THEOEEM. 

The  surface  of  a  sphere  is  to  tlte  ichole  surface  of  the  circum- 
scribed cylinder,  including  its  bases,  as  2  is  to  S  :  and  tJie 
solidities  of  these  two  bodies  are  to  each  other  in  the  same 
ratio. 


Let  MPXQ  be  a  great  circle  of  the  sphere;  ABCD  the 


BOOK    YIII. 


223 


circumscribed  square ;  if  the  semi- 
circle PMQ  and  the  half  square 
PADQ  are  at  the  same  time  made 
to  revolve  about  the  diameter  PQ^ 
the  semicircle  Avill  generate  the 
sphere,  while  the  half  square  will 
generate  the  cylinder  circumscribed 
about  that  sphere. 

The  altitude  AD  of  the  cylinder 
is  equal   to   the    diameter  PQ ;    the 

base  of  the  cylinder  is  equal  to  a  great  circle,  since  its 
diameter  AB  is  equal  to  J/xV;  hence,  the  convex  surface 
of  the  cylinder  is  equal  to  the  circumference  of  the  great 
circle  multiplied  by  its  diameter  (p.  1).  This  measure  is 
the  same  as  that  of  the  surfoce  of  the  sphere  (p.  10); 
hence,  tlie  surface  of  tlte  sphere  is  equal  to  the  convex  surface 
of  the  circumscribed  cylinder. 

But  the  surface  of  the  sphere  is  equal  to  four  great 
circles ;  hence,  the  convex  surface  of  the  cylinder  is  also 
equal  to  four  great  circles :  and  adding  the  two  bases,  each 
equal  to  a  great  circle,  the  total  surface  of  the  circumscrib 
ed  c^dinder  is  equal  to  six  great  circles ;  hence,  the  surface 
of  the  sphere  is  to  the  total  surface  of  the  circumscribed 
cylinder,  as  4  is  to  6,  or  as  2  is  to  3  ;  which  is  the  first 
branch  of  the  proposition. 

In  the  next  place,  since  the  base  of  the  circumscribed 
cylinder  is  equal  to  a  great  circle  of  the  sphere,  and  its 
altitude  to  the  diameter,  the  solidity  of  the  cylinder  is 
equal  to  a  great  circle  multiplied  by  its  diameter  (p.  2). 
But  the  solidity  of  the  sphere  is  equal  to  four  great  circles 
multiplied  by  a  third  of  the  radius  (p.  14) ;  in  other  terms, 
to  one  great  circle  multiplied  by  ^  of  the  radius,  or  by  I 
of  the  diameter;  hence,  the  sphere  is  to  the  circumscribed 
cylinder  as  2  to  3,  and  consequently,  the  solidities  of  these 
two  bodies  are  as  their  surfaces. 


Scholium  1.  Conceive  a  polyedron,  all  of  whose  fiiccs 
touch  the  sphere;  this  polj-edron  may  be  considered  as 
composed  of  pyramids,  each  pvramid  liaving  for  its  vertex 
the  centre  of  the  sphere,  and  for  its  base  one  of  the  poly- 


224r 


GEOMETRY 


edron's  fliccs.  Now,  it  is  evident  tliat  all  these  pyramids 
have  the  radius  of  the  spliere  for  their  common  alti- 
tude :  so  that  the  solidity  of  each  pyramid  will  be  equal 
to  one  face  of  the  polyedron  multiplied  by  a  third  of 
the  radius  :  lience,  the  whole  polj-edron  is  equal  to  its 
surf\ce  muhiplied  b}^  a  third  of  the  radius  of  the  inscrib- 
ed sphere. 

It  is  therefore  manifest,  that  the  solidities  of  polye- 
drons  circumscribed  about  the  sphere,  are  to  each  other 
as  their  surfaces.  Thus,  the  propert}^,  which  we  have 
shown  to  be  true  with  regard  to  the  circumscribed  cylin- 
der, is  also  true  with  regard  to  an  infinite  number  of 
other    solids. 

We  minht  likewise  have  observed,  that  the  surfaces  of 
polygons,  circumscribed  about  a  circle,  are  to  each  other 
as  their  perimeters. 


PEOrOSITIOX  XVI.      THEOREM. 


If  a  circular  segment  is  revolved  about  a  diameter  exterior  to 
it,  the  solid  generated  is  measured  by  one-sixth  of  ir  into 
tJie  square  of  the  chord,  into  the  distance  between  two  per- 
X>endiculars  let  fall  from  the  extremities  of  tlce  arc  on  the 
axis. 

Let    DXfB    be   a   circular   segment,    and    AC   the   axis 
about  which  it  is  revolved. 

On  the  axis,  let  fall  the  perpendic- 
ulars BE,  IJF  \  from  the  centre  (7, 
draw  CI  perpendicular  to  the  chord 
BD  ;    also  draw  the  radii   CB,  CD. 

The  solid  generated  by  the  sector 
CD  MB  is  measured  by  ^rrxCT^'xEF 
(p.  l-i,  s.  1).  The  solid  generated  by 
the  isosceles  triangle  CDB  has  for 
its  measure  l^xCl'^EF  (p.  12,  c.) ;  hence,  the  solid  gen- 
erated by  the  segment  DMB,  is  measured  by 


BOOK    VIII. 


225 


But  in  tlie  riglit-angled  triangle  CBI^  we  have  (b.  iy.  p.  8,  c.)^ 


CB 


cr=Br=\BD' 


hence,  the  solid  generated    by  the  segment  DMB^    has   for 
its  measure 

l-xEFx\Brf=l'r^xmrxEF. 

Scholium,    The  solid  generated   by  the  segment  BMD  is 
to  the  sphere  which  has  BD  for  a  diameter, 

as  I'jtxBD'xBF  is  to  \nfXBff,  or  as  EF  to  BD. 


PKOrOSITIOX  XVI T.     THEOREM. 

Every  segment  of  a  sphere  is  measured  hy  half  the  sum  of  its 
bases  mrdO'pU'ed  hy  its  altitude,  plus  the  solidity  of  a  spJiere 
whose  diameter  is  this  same  altitude. 

Let  DMB  be  the  arc  of  a  circle,  and  DF,  BE,  per- 
pendiculars let  fall  on  the  radius  CA  :  then,  if  the  area 
FBMBE  be  revolved  about  the  radius  CA  it  will  generate 
a  spherical  segment.  It  is  required  to  find  the  measure  of 
this  segment. 

The  solid  generated  by  the  circular 
segment  B3IB  is  measured  by  (p.  16) 

i'^xBlfxEF: 

the  frustum  of  the  cone  described  by 
the  trapezoid  FDBE  is  measured  bv 
(p.  6)  _      __ 

irfXEFxilU^r+VF^'+BExDE) : 
hence,    the   segment  of  the   sphere,  which   is   the   sum   of 
these  two  solids,  is  measured  by 

^':^XEF  x{2BE"+2UF'^+2BExDF+Blr). 
But  by  drawing  BO.  parallel  to  EF,  we  have, 

DO=DF-BE  and  lKr=IJE'-2DFxBE+BK  -, 

and,    Mr=m^+J)ff=EF'-\-]JF''-2DFxBE+BE^. 

Substituting   this   value   for  BB"  in  the  expression  for   the 
solidity  of  the  segment,  we  have, 


226  GEOMETRY. 

equal  to  ^':rXEFx{3RtJ'-^3lJF''+EF')  ; 

an  exprc'ssion   wliicli  may  be  written  in   t^vo  parts,  viz., 

EFx{^^^^^f^^)  and  J,x£F^ 

and  these  parts  correspond  with  the  enunciation. 

Cor.  If  the  radius  of  either  base  is  nothing,  the  seg- 
ment becomes  a  spherical  segment  with  a  single  base ; 
hence,  anij  spherical  segment,  with  a  single  base,  is  equivalent 
to  half  the  cylinder  having  the  same  base  and  the  same  altitude^ 
plus  the  sphere  of  which  this  altitude  is  the  diameter. 

GENERAL    SCHOLIUMS. 

1.  Let  i?  be  the  radius  of  a  cylinder's  base,  H  its  alti- 
tude:    the  solidity  of  the   cylinder  is 


2.  Let   B  be  the  radius   of  a   cone's  base,  U  its   alti- 
tude:  the  solidity  of  the  cone  is 

'rXJR\iII=i':tXB\K 

3.  Let  A  and  B  be  the  radii  of  the  bases  of  a  frustum 
of  a  cone,  //  its  altitude  :    the  solidity  of  the  frustum  is 

irrxIIxiA'+B^'+AxB). 

4.  Let  B  be  the  radius   of  a   sphere ;   its   solidity  is 


JcXi?. 


5.  Let  it  be  the  radius  of  a  spherical  sector,  R  the 
altitude  of  a  zone,  which  forms  its  base :  the  solidity  of 
the  sector  is 

i-rXi^'x//. 

6.  Let  P  and  Q  be  the  two  bases  of  a  spherical  seg^ 
ment,  //  its  altitude:   the  solidity  of  the  segment  is 

^^XE+i-zxlf. 

7.  If  the  spherical  segment  has  but  one  baiTO,  its 
solidity  is  -JPx"i7+i^X^l 


BOOK    IX. 


SPHERICAL    GEOMETRY. 


DEFINITIONS. 

1,  A  Spherical  Triangle  is  a  portion  of  the  surface 
of  a  sphere,  bounded  by  three  arcs  of  great  circles. 

These  arcs  are  named  the  sides  of  the  triangle,  and 
each  is  less  than  a  semicircumference.  The  angles  Avhich 
the  planes  of  the  circles  make  with  each  other,  are  the 
angles  of  the  triangle. 

2.  A  spherical  triangle  takes  the  name  of  right-angled^ 
isosceles^  equilateral^  in  the  same  cases  as  a  rectilineal  tri- 
angle. 

8.  A  Spherical  Polygon  is  a  portion  of  the  surface 
of  a  sphere  bounded  by  three  or  more  arcs  of  great  circles. 

4.  A  LuNE  is  a  portion  of  the  surface  of  a  sphere  in- 
cluded between  two  semi-circles  intersecting  in  a  common 
diameter  of  the  sphere. 

5.  A  Spherical  Wedge,  or  Ungula,  is  that  portion 
of  a  solid  sphere,  included  between  two  planes  passing 
through  the  centre,  and  the  lune  which  forms  its  base. 

6.  A  Spherical  Pyramid  is  a  portion  of  the  solid 
sphere,  included  between  three  or  more  planes.  The  base 
of  the  pyramid  is  the  spherical  polygon  intercepted  by  the 
Bame  planes.  These  planes  bound  a  polyedral  angle,  whose 
vertex  is  at  the  centre  of  the  sphere. 

7.  The  Pole  of  a  Circle  is  a  point  on  the  surfxce  of 
the  sphere,  equally  distant  from  every  point  in  the  circuni- 
ference. 


228 


GEOMETRY. 


PEOrOSITION   I.     THEOREM. 

Tn  every  spherical  tnariQle^  any  side  is  less  than  the  sum  of  Oie 
two  oilier  sides. 

Let  0  be  the  centre  of  the  sphere,  and  ACB  ^  spheri- 
cal triangle  :  then  will  any  side  be  less  than  the  sum  of 
the  two  other   sides. 

For,  draw  the  radii  OA,  OB,  OC. 
Conceive  the  pLanes  AOB,  AOC,  COB, 
to  be  drawn  ;  these  planes  bound  a 
polyedral  angle  whose  vertex  is  at 
the  centre  0 ;  and  the  plane  angles 
AOB,  AOC,  COB,  are  measured  by 
AB,  AC,  BC,  the  sides  of  the  spheri- 
cal triangle.  But  each  of  the  three 
plane  angles  which  bound  a  polyedral 
angle  is  less  than  the  sum  of  the  two  other  angles  (b.  VI., 
P.  19) ;  hence,  any  side  of  a  spherical  triangle  is  less  than 
the  sum  of  the  two  other  sides. 


PKOrOSITION  11.    TIIEOKEM. 

The  sum  of  all  the  sides  of  any  spherical  polygon   is    less  than 
tlte  circumference  of  a  great  circle. 

Let  ABODE  be  any  sjiherical  polj'gon,  and  0  the  cen- 
tre of  the  sphere. 

Conceive  0  to  be  the  vertex 
of  a  polyedral  angle  bounded 
by  the  plane  angles  AOB,  BOC, 
COD,  &c.  Now,  the  sum  of  the 
plane  angles  which  bound  a  poly- 
edral angle  is  less  than  four 
right  angles  (b.  yi.,  p.  20) ;  hence, 
the  sum  of  the  sides  of  any 
spherical  polygon  is  less  than  the  circumference. 

Cor.  The  sum  of  the  three  sides  of  any  spherical  tn- 
angle  is  less  than  the  circumference ;  for,  the  triangle  is  a 
polygon  of  three  sides. 


BOOK  IX. 


229 


PROPOSITION   III.     THEOREM. 


The  poles  of  a  great  circle  of  a  sphere  are  the  exlremili'es  of 
that  diameter  of  the  S2)here  ichich  is  per^jeyKZ/ci/Zar  to  the. 
circle;  and  these  extremities  are  also  the  poles  of  all  small 
circles  parallel  to  it. 

Let  FD  be  perpendicular  to  the  great  circle  AJfB ;  then 
will  U  and  D  be  its  poles ;  and  they  will  also  be  the  poles 
of  every  parallel  small  circle  FNG. 

For,  DC  being  perpen- 
dicular to  the  plane  AMB^ 
is  perpendicular  to  all  the 
straight  lines  CA,  CM,  CB, 
&c.,  drawn  through  its  foot 
in  this  plane  (b.  vl,  d.  1) ; 
hence,  all  the  arcs  Z^J, 
jDJ/,  DB,  &c.,  are  quarters 
of  the  circumference.  So 
likewise  are  all  the  arcs 
FA,  FU,  FB,  kc. ;  there- 
fore, the  points  F  and  F 
are  each  equally  distant  from  all  the  points  of  the  circum- 
ference AMB  ]  hence,  they  are  the  poles  of  that  circum- 
ference (d.  7). 

Again,  the  radius  FC,  perpendicular  to  the  plane  A3fBy 
is  perpendicular  to  the  parallel  FXG  ;  hence,  it  passes 
thiough  0,  the  centre  of  the  circle  FNG  (b.  viii.,  P.  7,  c.  4); 
hence,  if  the  chords  FF,  FN,  FG,  be  drawn,  these  oblique 
lines  will  cut  off  equal  distances  measured  from  0 ;  hence, 
they  will  be  equal  (b.  vl,  p.  5).  But,  the  chords  being 
equal,  the  arcs  are  equal ;  hence,  the  point  D  is  the  pole 
of  the  small  circle  FNG  ;  and  for  like  reasons,  the  point 
E  is  the  other  pole. 

Cor.  If  through  the  pole  D  and  any  point  J/,  in  the  aro 
of  a  great  circle  A  MB,  an  arc  of  another  great  circle  JIB  be 
drawn,  the  arc  MD  is  a  quarter  of  the  circumference,  and  is 
called  a  quadrant.  This  quadrant  makes  a  right  angle  with 
the  arc  AM.  For,  the  line  FC  being  perpcndicnlar  to  the 
plane  AMC,  every  plane  DMF,  passing  thiough  the  line  DC  is 


230 


GEOMETRY. 


perpendiculaT  to  tlie  plane 
AMC  (b  vl,  p.  16);  hence, 
tlie  angle  of  these  planes, 
or  the  angle  AMD  is  a 
right  angle. 

Cor.  2.  Conversely :  If 
the  distance  of  the  point 
D  from  each  of  the  points 
A  and  JiJ  in  the  circum- 
ference of  a  great  circle, 
is  equal  to  a  quadrant,  the 
point  D  is  the  pole  of  the 
arc  All. 

For,  let  0  be  the  centre  of  the  sphere,  and  draw  the 
radii  CI),  CA,  CM.  Since  the  angles  ACD,  MCD,  are  right 
angles,  the  line  CD  is  perpendicular  to  the  two  straight 
lines  CA,  CM',  hence,  it  is  perpendicular  to  their  plane 
(b.  VI.,  p.  4) :  hence,  the  point  D  is  the  pole  of  the  arc  AM. 

Scholium.  The  properties  of  these  poles  enable  us  to 
describe  arcs  of  a  circle  on  the  surface  of  a  sphere,  with 
the  same  facility  as  on  a  plane  surface.  It  is  evident,  for 
instance,  that  by  turning  the  arc  DF,  or  any  other  line 
extending  to  the  same  distance,  round  the  point  D,  the 
extremity  F  will  describe  the  small  circle  FNG ;  and  by 
turning  the  quadrant  DFA  round  the  point  D,  its  extrem- 
ity A  will  describe  the  arc  of  a  great  circle  AMB. 


PEOPOSITION  IV.     TIIEOKEM. 

The  angle  formed  hy  two  arcs  of  great  circles,  is  equal  to  the 
angle  formed  hy  the  tangents  of  these  arcs  at  their  point 
of  intersection.  The  angle  is  measured  hy  the  arc  of  a 
great  circle  described  from  the  vertex  as  a  pole,  and  limiied 
hy  the  sides,  j^f^oduced  if  necessary. 

Let  the  angle  BAC  be  formed  by  the  two  arcs  AB^ 
AC',  then  will  it  be  equal  to  the  angle  FAG  formed  by 
the  tangents  AF,  AG,  and  be  measured  by  the  arc  DE  of 
a  great  circle,  described  about  J.  as  a  pole. 


BOOK    IX, 


231 


For,  tlie  tangent  AF,  drawn  in  the 
plane  of  tlie  arc  AB,  is  perpendicular  to 
the  radius  A  0 ;  and  the  tangent  A  G, 
drawn  in  the  plane  of  the  arc  ACj  is 
periDcndicular  to  the  same  radius  AO. 
Hence,  the  angle  FAG  is  equal  to  the 
angle  contained  bj  the  planes  ABDII, 
ACFII  {b.yi.,I).  4:)]  Avhich  is  that  of 
the  arcs  AB,  AC,  and  is  called  the  angle 
BAG. 

Again,  if  the   arcs  AI)  and  AF  are 
both  quadrants,  the  lines  OD,  OF,  are  perpendicular  to  OA, 
and   the   angle  BOF  is  eqjial  to  the   angle  of   the   planes 
ABBH,  ACFH;   hence,  the  arc  BF  is  the  measure  of  the 
angle  contained  by  these  planes,  or  of  the  angle  CAB. 

Cor.  1.  The  angles  of  spherical  triangles  ma}^  be  com- 
pared together,  by  means  of  the  arcs  of  great  circles  des- 
cribed from  their  vertices  as  poles  and  included  between 
their  sides:  hence,  it  is  easy  to  make  an  angle  of  this 
kind  equal  to  a  given  angle. 

Cor.  2.  Vertical  angles,  such 
as  ACQ  and  BCJ^  are  equal ;  for 
either  of  them  is  still  the  angle 
formed  by  the  two  planes  ACBj 
OCK 

It  is  further  evident,  that,  when 
two  arcs  ACB,  OCN,  intersect, 
the  two  adjacent  angles  ACQ, 
OCB,  taken  together,  are  equal 
to  two  ri":ht  ano-les. 


PROPOSITION  v.     THEOREM. 


Tf  from  the  vertices  of  the  three  angles  of  a  spherical  triangle^ 
as  poles,  arcs  he  described  forming  a  spherical  triangle; 
then,  the  vertices  of  the  angles  of  this  second  triangle,  loill 
he  respectively  poles  of  tlie  sides  of  the  first. 

From   the   vertices  A,  B,   C,  as  poles,  let  the   arcs  FF, 
FDy  FB,  be  described,  forming  on  the  surface  of  the  sphere, 


232 


GEO^rETRY. 


the  triaiigle  BFE \  then  will  the  vertices  Z),  E^  and  jP,  be 
respectively  poles  of  the  sides  i>(7,  AC^  AB. 

For,  the  point  A  being 
the  pole  of  the  arc  EF^  the 
distance  ^•i£'  is  a  quadrant; 
the  point  C  being  the  pole 
of  the  arc  BE,  the  distance 
CE  is  likewise  a  quadrant: 
hence,  the  point  E  is  re- 
moved the  length  of  a  quad- 
rant from  each  of  the  points 
A  and  C\    hence,  it  is  the 

pole  of  the  arc  AG  (p.  3,  c.  2).  It  may  be  shown  by  simi- 
lar reasoning,  that  B  is  the  pole  of  the  arc  BC^  and  F 
that  of  the  arc  AB. 

Scholium.  Hence,  the  triangle  ABC  may  be  described 
by  means  of  BEF,  as  BEE  is  described  by  means  of  ABC. 
Triangles  so  described,  are  called  polar  trlanglts^  or  siipple- 
mental  triangles. 


PROPOSITION  VI.     TIIEOKEil. 


The  same  siqiposiiion  continuing  as  in  the  last  Proj^sifiorij 
each  angle  in  one  of  iJie  triangles,  tvill  he  measured  hy  a 
semicircumference,  minus  tlie  side  lying  opptosite  to  it  in  tht 
other  triangle. 

For,  produce  the  sides 
AB,  AC,  if  necessary,  till 
they  meet  EF,  in  G  and  //. 
The  point  A  being  the  pole 
of  the  arc  GH,  the  angle 
A  is  measured  by  that  arc 
(p.  4).  But,  since  E  is  the 
pole  of  xill,  the  arc  EH  is 
a  quadrant ;  and  since  F  is 
the  pole  of  AG,  EG  is  a 
quadrant:  hence,  EH+GF  is  equal  to  a  semicircumference. 
But,  EH-\-GF=EF+GII\  hence  the  arc  GR,  which  moa- 


BOOK    IX. 


233 


Bures  the   angle  A,  is  equal   to   a   semicircumference  mmiis 
the    side   UF.      In  like    manner,  the  angle  B  is   measured 


byi 


circ- 


DF:    the  angle   C,  by  \circ-DE, 


This  property  is  reciprocal  in  the  two  triangles,  since 
each  of  tlieni  is  described  in  a  similar  manner  by  means 
of  the  other.  Thus  the  angle  1\  for  example,  of  the  tri- 
angle EOF,  is  measured  by  the  arc  Ml  \  but  MI+BC= 
MC^-B[=\circ.\  hence,  the  arc  J/TJ  the  measure  of  Z),  is 
equal  to  \drc.  —  BC:  the  angle  E  is  measured  by  \cLrc.—AG^ 
and  the  angle  E  by  \cLrc.—AB. 

Scholium.  It  must  farther  be  ob- 
served, that  besides  the  triangle  DEF^ 
three  others  might  be  formed  by  the 
intersection  of  the  three  arcs  JJE^ 
EF^  DF.  But  the  proposition  is  ap-  -^^ 
plicable  only  to  tlie  central  triangle, 
Avhich  is  distinguished  from  the  other 
three  by  the  circumstance,  that  the  two  angles  A  and  D 
lie  on  the  same  side  of  BC,  the  two  B  and  E  on  the  same 
side  of  AC,  and  the  two  0  and  F  on  the  same  side  of  AB. 


rKorosiTioN  vii.    theorem. 

If  aronnd  the  vertices  of  any  two  angles  of  a  given  qiHierical 
triangle,  as  poles,  the  circumferences  of  two  circles  he  des- 
cribed which  shall  pass  tJirough  the  vertex  of  the  third  an- 
gle of  the  triangle:  if  then,  through  the  other  point  in  ichick 
these  circumferences  intersect  and  the  vertices  of  the  first  two 
angles  of  the  triangle,  two  arcs  of  great  circles  he  drawn, 
the  triangle  thus  formed  icill  have  all  its  p)arts  equal  to 
those  of  the  given  triangle,  each  to  each. 

Let  ABC  be  the  given  triangle,  CED,  DEC,  the  arcs 
described  about  A  and  B  as  poles  ;  then  will  the  triangles 
ABC,  ABB  have  all  their  parts  equal  each  to  each. 

For,  by  construction,  the  side  AB=AC,  DB=BC,  and 
AB  is  common ;  hence,  these  two  triangles  have  their  sides 
equal,  each  to  each.  We  are  now  to  show,  that  the  argles 
opposite  these  equal  sides  are  also  equal,  each  to  each. 


23-i 


GEOMETRY. 


If  the  centre  of  the  sphere  is 
at  0,  a  triedral  angle  may  be  con- 
ceived as  formed  at  0  by  the  three 
plane  angles  A  OB,  AGO,  BOO; 
likewise  another  triedral  angle  may 
be  conceived  as  formed  by  the 
three  plane  angles  AOB,  AOD^ 
BOD.  And,  because  the  sides  of 
the  triangle  ABC  are  equal  to 
those  of  the  triangle  ABB,  the  plane  angles  forming  the 
one  of  these  triedral  angles,  are  equal  to  the  plane  angles 
forming  the  other,  each  to  each  :  hence,  the  planes  are 
equally  inclined  to  each  other  (b.  yi.,  p.  21) ;  and  all  the 
angles  of  the  spherical  triangle  DAB^  are  respectively 
equal  to  those  of  the  triangle  CAB,  namely,  I)AB=BAC, 
DBA=ABC,  and  AI)B=ACB;  consequently,  the  sides  and 
the  angles  of  the  triangle  ABB,  are  equal  to  the  sides  and 
the  angles  of  the  triangle  ACB,  each  to  each. 

Scholium.  The  equality  of  these  triangles  is  not,  how- 
ever, an  absolute  equality,  or  one  of  superposition  :  for,  it 
would  be  impossible  to  apply  them  to  each  other,  unless 
they  were  isosceles.  The  equality  meant  here  is  what  we 
have  already  named  an  equality  by  symmetry  (b.  yi.,  21,  s.  3) ; 
therefore,  we  shall  call  the  triangles  ACB,  ABB,  symmetri- 
cal triangles. 

PKOPOSITIOX    YIII.     TIIEOEEM. 


Two   trianrjles    on    the    same   spliere,    or   on  equal  spheres^    are 

equal  in  all  their  parts,  ichen   two  sides   and  the  included 

angle  of  the  one  are  equcd   to   two   sides   and  the  included 
angle  of  the  other,  each  to  each. 


Let  ABC,  EFG,  be  two  trian- 
gles having  the  side  AB=FF, 
the  side  AC=EG,  and  the  angle 
BAC=FEG;  then  will  the  two 
triangles  be  equal  in  all  their 
parts. 

For,  the  triangle  EFG  may  be 
placed  on  the  triangle  ABC,  or  on 


C  G 


BOOK    IX. 


235 


ABD  symmetrical  with  ABC^  just  as  two  rectilineal  trian- 
gles are  placed  upon  each  other,  when  they  have  an  equal 
angle  included  between  equal  sides.  Hence,  all  the  parts 
of  the  triangle  EFG  are  equal  to  all  the  parts  of  the  tri- 
angle ABC  \  that  is,  besides  the  three  parts  equal  by 
hypothesis,  we  have  the  side  BC—FG^  the  angle  ABC=^ 
EFG,  and  the  angle  ACB=EGF. 


PROPOSITION  IX.     THEOREM. 

Two  triangles  on  the  same  sphere  or  on  equal  sj^heres^  are  equal 
in  all  their  qjarts,  tohen  two  angles  and  the  included  side 
of  the  one  are  equal  to  two  angles  and  the  included  side  of 
the  other,  each  to  each. 

For,  one  of  these  triangles,  or  the  triangle  symmetrical 
with  it,  may  be  placed  on  the  other,  as  is  done  in  the 
corresponding  case  of  rectilineal  triangles  (b.  i.,  p.  6). 


PROPOSITION   X.     THEOREM. 

Jf  two  triangles  on  the  same  sphere,  or  on  equcd  spheres,  have 
all  their  sides  equal,  each  to  each,  their  angles  ivill  likewise 
he  equal,  each  to  each,  the  equal  angles  hjing  op)posite  the 
equcd  sides. 


The  truth  of  this  proposition  is  evi- 
dent from  Prop.  YIL,  where  it  was 
shown,  that  with  three  given  sides  AB, 
AG,  BC,  only  two  triangles  ACB,  ABD, 
can  be  constructed,  and  that  these  tri- 
angles will  have  all  their  parts  equal 
each  to  each.  Hence,  the  two  trian- 
gles, having  all  their  sides  respectively 
equal,  must  either  be  absolutely  equal, 
or  symmetrically  equal;  in  either  of  which  cases,  their  cor- 
responding angles  are  equal,  and  lie  opposite  to  equal 
sides. 


236  GEOMETRY, 


TEOPOSITION    XI.     THEOEEM. 

In  every  isoscdes  spherical  triangle,  the  angles  opposite  the  equal 
sides  are  equal;  and  conversely,  if  two  angles  of  a  spherical 
triangle  are  equal,  the  triangle  is  isosceles. 

First.  Suppose  tlie  side  AB=AC]  ^ve  sliall  Lave  the 
angle  C=B. 

■  For,  if  the  arc  AD  be  dra\vn  from 
the  vertex  A  to  the  middle  point  B  of 
the  base,  the  two  triangles  ABD,  ACD, 
will  have  all  the  sides  of  the  one  res- 
pectively equal  to  the  corresponding  sides 
of  the  other,  viz.,  AB  common,  BB=BC, 
and  AB=AO :  hence,  by  the  last  propo- 
sition, their  angles  will  be  equal;    therefore,  B=C. 

Secondly.  Suppose  the  angle  B  —  C\  we  shall  have  the 
side  AC=AB. 

For,  if  not,  let  AB  be  the  greater  of  the  two ;  take 
BO=AC,  and  draw  OC.  Then,  in  the  two  triangles  BOO, 
BAC,  the  two  sides  BO,  BC,  are  equal  to  the  two  AC,  BC; 
the  angle  OBC,  contained  by  the  first  two  is  equal  to  A  CB 
contained  by  the  second  two.  Hence,  the  two  tiiangles 
BOC,  ACB,  have  all  their  other  parts  equal  (p.  8) ;  hence, 
the  angle  OCB—ABC:  but,  by  hypotliesis,  the  angle 
ABC = ACB;  hence,  we  have  OCB=ACB,  which  is  absurd 
(a.  8);  therefore,  an  absurdity  follows -if  vre  suppose  AB 
different  from  AC;  hence,  the  sides  AB,  AC,  opposite  to 
the  equal  angles  B  and   C,  are  equal. 

Scholium.  Since*  the  triangles  BAD,  DAC,  are  equal  in 
all  their  parts  (p.  10),  the  angle  BAD=DAC,  and  BDA=- 
ADC:  consequently,  ADB  and  ADC,  are  right  angles: 
hence,  the  arc  drawn  from  the  vertex  of  an  isosceles  spherical 
triangle  to  the-  middle  of  the  base,  is  at  right  ang^^  to  tJie  base 
and  bisects  the  vertical  angle. 


BOOK    IX. 


237 


TEOrOSITIOX   XII.     TIIEOEEM. 


hi  any  sjjJierical  trianrjle,  the  greater  side  is  opposite  the  greater 
angle;  and  conversely^  the  greater  angle  is  opposite  the 
greater  side. 

Let  the  angle  A  be  greater  than  the  angle  i?,  then  will 
BC  be  greater  than  AC]  and  conversely,  if  BC  is  greater 
than  AC^  then  will  the  angle  A  be  greater  than  B. 

First.  Supjoose  the  angle 
AyB]  make  the  angle  BAD 
=B ;  then  we  shall  have 
AD=DB  (p.  11)  ;  but  AD-^ 
DC  is  gi^eater  than  AC]  hence, 
putting  DB  in  place  of  AD^ 
we  shall  have  DB+DC>AC,  or  BCyAC. 

Secondly.  If  we  suppose  BCyAC^  the  angle  BAC  \\\\\ 
be  greater  than  ABC.  For,  if  BAC  were  equal  to  ABC^ 
we  should  have  BC=AC]  if  BAC  were  less  than  ABC^ 
we  should  then,  as  has  just  been  shown,  find  BC<CAC. 
Either  of  these  conditions  is  contrary  to  the  supposition : 
hence,  the  angle  BAC  is  greater  than  ABC. 


TEOrOSITION  XIII.     TIIEOEEM. 


J^  two  triangles    on    the   same  sphere^  or  on  equal  spheres^   arc 
mutually  equiangular^  they  are  cdso  mutually  equilateral. 

Let   A   and  B  be    the    two    given    triangles ;   P  and  Q 
their  polar  triangles. 

Since  the  angles  are  equal,  each  to  each, 
in  the  triangles  A  and  i>,  the  sides  are  equal 
each  to  each,  in  their  polar  triangles  P  and  Q 
(p.  6) :  but,  since  the  triangles  P  and  Q  are 
mutually  equilateral,  they  must  also  be  mutu- 
ally equiangular  (p.  10) ;  and  lastly,  the  an- 
gles being  equal,  each  to  each,  in  the  triangles 
P  and  Q^  it  follows  that  the  sides  arc  equal 
each  to  each,  in  their  polar  triangles  A  and  B. 


larr:^ 


238  GEOMETRY. 

Hence,  the   mutually  equiangular  triangles  A  and  B  are  at 
the  same  time,  mutually  equilateral. 

ScJioJium.  This  proposition  is  not  applicable  to  recti- 
lineal triangles;  in  which  equality  among  the  angles  indi- 
cates only  proportionality  among  the  sides.  Nor  is  it  diffi- 
cult to  account  for  the  difference,  in  this  respect,  between 
spherical  and  rectilineal  triangles.  In  the  proposition  now 
before  us,  as  well  as  in  the  preceding  ones,  which  treat 
of  the  comparison  of  triangles,  it  is  expressly  required  that 
the  arcs  be  traced  on  the  same  sphere,  or  on  equal  spheres. 
N'ow,  similar  arcs  are  to  each  other  as  their  radii;  hence, 
on  equal  spheres,  two  triangles  cannot  be  similar  without 
being  equal.  Therefore,  it  is  not  strange  that  equality 
among  the  angles  should  produce  equality  among  the 
sides. 

The  case  would  be  different,  if  the  triangles  were  drawn 
upon  unequal  spheres  ;  there,  the  angles  being  equal,  the 
triangles  would  be  similar,  and  the  homologous  sides  would 
be  to  each  other  as  the  radii  of  their  spheres. 


PEOPOSITIOX   XIV.     TREOEEM. 

The  sum  of  all  the  angles^  in  any  splierical  triangle,  is  less  than 
six  right  angles  and  greater  than  two. 

For,  in  the  first  place,  every  angle  of  a  spherical  trian- 
gle is  less  than  two  right  angles :  hence,  the  sum  of  the 
three  is  less  than  six  right  angles. 

Secondly,  the  measure  of  each  angle  of  a  spherical  trian- 
gle is  equal  to  the  semicircumference  minus  the  correspond- 
ing side  of  the  polar  triangle  (p.  6)  ;  hence,  the  sum  of 
the  three,  is  measured  by  the  three  semicircumferences, 
'inimis  the  sum  of  the  sides  of  the  polar  triangle.  Kow, 
this  latter  sum  is  less  than  a  circumference  (p.  2,  c.) ;  there- 
fore, taking  it  away  from  three  semicircumferences,  the 
remainder  is  greater  than  one  semicircumference,  which 
is  the  measure  of  two  right  angles;  hence,  the  sum  of  the 
three  angles  of  a  spherical  triangle  is  greater  than  two 
right  angles. 


BOOK   IX, 


239 


Cot.  1.  The  sum  of  tlie  three  angles  of  a  spherical  tri- 
angle is  not  constant,  like  that  of  the  angles  of  a  recti- 
lineal triangle,  but  varies  between  two  right  angles  and 
six,  without  ever  reaching  either  of  these  limits.  Two 
given  angles  therefore  do  not  serve  to  determine  the  third. 

Cor.  2.  A  spherical  triangle  may  have  two,  or  even 
three  of  its  angles  right  angles ;  also  two,  or  even  three 
of  its  angles  obtuse. 

Cor.  3.  If  the  triangle  ABC  is  U-recian- 
gular,  in  other  words,  has  two  right  angles 
B  and  C,  the  vertex  A  is  the  pole  of  the 
base  BC ;  and  the  sides  AB,  A  C,  are  quad- 
rants (p.  3,  c.  2). 

If  the  angle  A  is  also  a  right  angle,  the 
triangle  ABC  is  iri-rectangular ;  each  of  its  angles  is  a  right 
angle,  and  its  sides  are  quadrants.  Two  tri-rectangular  tri- 
angles make  half  a  hemisphere,  four  make  a  hemisphere, 
and  eight  the  entire  surljice  of  a  sphere. 


PKOPOSITION  XV.     THEOKEM. 

The  surface  of  a  lime  is  to  the  surface  of  the  s])here,  as  the 
angle  of  the  lune,  to  fii<r  rigid  angles ;  or^  as  the  arc 
which  measures  that  angle,  to  the  circumference. 

Let  AMBN  be  a  lune,  and  NCM  the  angle  included 
between  its  two  great  circles :  then  Avill  its  surface  be  to  the 
surface  of  the  sphere  as  the  angle  NCM  to  four  right  angles, 
or  as  the  arc  NM  to   the  circumference  of  a  great  circle. 

For,  suppose  the  arc  AfN  to  be 
to  the  circumference  MNPQ,  as  some 
one  integer  number  to  another,  as 
5  to  48,  for  example.  Divide  the 
circumference  MNPQ,  into  48  equal 
parts,  AfN  will  contain  5  of  them , 
and  if  the  pole  A  were  joined  with 
the  several  points  of  division,  by  as 
many  quadrants,  we  should  in  the 
hemisphere  A  MNPQ,  have  48  triangles,  all  equal,  because 
all  the  corresponding  parts  are  equal.     The  whole    sphere 


2i0  GEO^LETKY. 

would  contain  96  of  tlicse  triangles,  and  tbe  lune  AMBNA^ 
10  of  them  ;  hence,  the  kine  is  to  the  spliere  as  10  is  to 
96,  or  as  5  to  48  ;  in  otiier  words,  as  the  arc  MX  is  to 
the  circunifereisce. 

If  the  arc  MX  is  not  commensurable  with  the  circum- 
ference, it  niav  still  he  shown,  that  the  lune  is  to  the 
sphere  as  MX  to  the  circumference  (b.  ill.,  P.  17). 

Cor.  1.  Two  lunes  on  the  same  or  on  equal  spheres, 
are  to  each  other  as  their  respective  angles. 

Cor.  2.  It  was  shown  above,  that  the  Avhole  surface  of 
the  sphere  is  equal  to  eight  tri-rectangular  triangles  (p.  14, 
C.  3) ;  hence,  if  the  area  of  one  such  triangle  be  represent- 
ed by  T^  the  surface  of  the  whole  sphere  Avill  be  express- 
ed by  ^T.  This  granted,  if  the  right  angle  be  assumed 
equal  to  1,  the  surface  of  the  lune  whose  angle  is  A^  will 
be  expressed  by  lAxT.     For, 

4     :     ^1     :  :     82^    :     2.4x^, 
in  which  expression,  A  represents  such  a  part  of  unity,  as 
the  angle  of  the  lune  is  of  one  right  angle. 

Scholium.  The  spherical  ungula,  bounded  by  the  planes 
AM  By  AXJJ,  is  to  the  whole  solid  sphere,  as  the  angle  A 
is  to  four  right  angles.  For,  the  lunes  being  equal,  the 
spherical  ungulas  are  also  equal ;  hence,  two  spherical 
ungulas  are  to  each  other,  as  the  angles  formed  by  the 
planes  which  bound  them. 

PEOrOSlTIoN    XVI.     TIIEOKEM. 
Tico  symmetrical  sjJicrical  trianrjles  are  equivalent. 

Let  ABCy  DEF^  be  two  s3-mmetrical  triangles,  that  is 
to  say,  two  triangles  having  their  sides  AB=I)L]  AC=DF, 
CB=EFy  and  yet  incapable  of  superposition:  we  are  to 
show  that  the  surface  ABC  is  equal  to  the  surface  DEF. 

Let  P  be  the  pole  of  the  small  circle  passing  through 
the  three  points  J,  Z>,  C  f'  from  this  point  draw  the  equal 

*  The  circle  wliich  passes  through  the  tlirce  points  A^  B,  C,  or  which  circum- 
Bca-ibes  the  triaiiirle  ABC,  am  only  be  a  t^iiiall  circle  of  the  sphere ;  for  if  it  were 
a  great  circle,  the  three  sides,  AB,  BC\  AC,  would  he  iu  cue  plane,  and  the  tri- 
tuigla  ABC  would  be  reduced  to  one  of  ita  sides. 


BOOK    IX. 


24) 


D 

4\ 


>Q  Pv 


"E 


--;0 


arcs  PA,  PB,  PC  (p.  3);  at  the 
point  F  make  t'^e  angle  1jFQ= 
ACP,  the  arcFQ=CP;  and  draw  / 

DQ,PQ. 

The  sides  JJF^  FQ^  are  equal 
to  the  sides  AC\  CP  \  the  angle 
DFQ=A  CF ;  hence,  the  two  tri- 
angles PFQ,  A  Cl\  are  equal  in 
all  their  parts  (p.  8)  ;  consequently,  the  side  DQ—APy 
and  the  angle  DQF--^APa 

In  tlie  tri:uiglc3  JJFE,  ABC,  the  angles  PFF,  ACB, 
opposite  to  tlje  equrd  sides  JJE,  AB,  are  equal  (r.  10).  K 
the  angles  PFQ,  ACP,  which  are  equal  by  construction, 
be  taken  away  from  them,  there  will  remain  the  angle 
QFE,  equal  to  PCB.  The  sides  QF,  FE,  are  equal  to  the 
sides  PC,  CB ;  hence,  the  two  triangles  FQE,  CPB,  are 
equal  in  all  their  parts  (r.  8) ;  hence,  the  side  QE=PB, 
and  the  angle   FQE=^CPB. 

Now,  the  triangles  DFQ,  ACP,  wbicli  have  their  sides 
respectively  equal,  are  at  the  same  time  isosceles,  and  capa- 
ble of  coinciding,  when  applied  the  one  to  the  other. 
For,  having  placed  AC  on  its  equal  DF,  the  equal  sides 
will  fall  the  one  on  the  other,  and  thus  the  two  triangles 
will  exactly  coine.'de:  hence,  the}^  are  equal;  and  the  sur- 
face DQF  —  APC  For  a  like  reason,  the  surfiicc  FQE  = 
CPB,  and  the  surfac-.  BQE=APB',    hence  we  have, 

BQF+FQE-DQE^Z^APC+CPB-APB, 
or,  DFE=o=ABC', 

lience,  the  two  sj^mmetrical  triangles  ABC,  DEF,  arc  equal 
in  surf\ice. 

SclioUam.  The  poles  P  and  Q  might  lie  within  triangles 
ABC,  DEF:  in  which  case  it  would  be  requisite  to  add 
the  three  triangles  DQF,  FQF,  BQE,  together,  in  order  to 
make  up  the  triangle  BEE',  and  in  like  manner,  to  add 
the  three  triangles  APC,  CPB,  APB,  together,  in  order  to 
make  up  the  triangle  ABC',  -in  all  other  respects, '  the 
demonstration  and  the  result  would  be  the  same. 


IG 


242 


GEOMETRY, 


PROPOSITION  XVII.      TIIEOKEM. 

IJ  the  circumferences  of  two  great  circles  intersect  each  n/ier  on 
tJie  surface  of  a  hemisphere,  tlce  sura  of  the  opposite  trian- 
gles thus  formed^  is  equivalent  to  tlie  surface  of  a  lune 
u-Jiose  angle  is  equal  to  the  angle  formed  hg  tlie  circles. 

Let  the  circumferences  A  OB,  COB^  intersect  on  the  sur- 
fcice  of  a  hemisphere;  then  Avill  the  opposite  triangles 
AOC^  BOD^  be  equivalent  to  the  lune  whose  angle  is  BOB. 

For,  produce  the  arcs  OB^  OB^  on 
the  other  hemisphere,  till  they  meet 
in  iV.  Now,  since  AOB  and  OBX 
are  semicircumferences,  if  we  take 
away  the  common  part  OB,  Ave  shall 
have  BN=AO.  For  a  like  reason, 
we  have  BN=CO,  and  BB-^AG. 
Hence,  the  two  triangles  AOC,  BBN^ 
have  their  three  sides  respectively  equal  :  they  are  there- 
fore symmetrical;  hence,  they  are  equal  in  surface  (p.  16). 
But  the  sum  of  the  triangles  BDN,  BOB^  is  equivalent  to 
the  lune  OBXBO,  whose  angle  is  BOB  :  hence,  AOO+BOB 
IS  equivalent  to  the  lune  whose  angle  is  BOB. 

Scholium.  It  is  likewise  evident,  that  the  two  spherical 
pyramid^,  which  have  the  triangles  AOC,  BOB,  for  bases, 
are  together  equivalent  to  the  spherical  ungula  whose  angle 
is  BOB. 


PKOPOSITIOX   XVIII.     THEOEEM. 

The  surface  of  a  spherical  triangle  is  equal  to  the  excess  of  the 
sum  of  its  three  angles  above  two  rigid  angles,  multip)lied 
hy  the  tri-rectangular  triangle. 

Let  ABC  be  any  spherical  triangle :  then  will  its  sur- 
face be  equal  to 

{A-\-B+C-2)xT. 

For,  produce  its  sides  till  they  meet  the  great  circle 
DEFG,  drawn  at  pleasure,  without  the  triangle.  By  the 
last   theorem,  the   two   triangles  ABE,  AGH,  are   together 


BOOK    IX.  243 

equivalent  to  the   lane   whose   angle   is 

A,  and   which    is    measured    b}^  2AxT 

(p.  15,  c,  2).      Hence,    Ave   have  ADE  + 

AGIJ=2AxT;    and,  for  a  like   reason, 

BGF-\-BW==2Bx7]    and    CIII+CFB 

^2CxT.      But    the    sum  of  these    six 

triangles    exceeds    the    liemispheie     by 

twice  the  triangle  A  BC,  and  the  hemisphere  is  represented 

by  d7':   therefore,  twice  the  triangle  ABC,  is  eqaivalcut  to 

2AxT+2BxT-{-2CxT-4:T] 
and,  consequently^, 

ABC^o{A+B  +  C-2)xT', 
hence,  every  spherical  triangle  is  measured   by  the  sum  of 
its  three  angles  minus   two    right  angles,  multiplied  by  the 
tri-rectangular  triangle. 

Scholium  1.  When  we  speak  of  the  spherical  angles,  we 
regard  the  right  angle  as  unit}^,  and  compare  the  sum  of 
the  three  angles  with  this  standard.  Hence,  however 
many  right  angles  there  may  be  in  the  sum  of  the  three 
angles  minus  two  right  angles,  just  so  many  tri-rectangular 
triangles,  will  the  proposed  triangle  contain.  If  the  angles, 
for  example,  are  each  equal  to  f  of  a  right  angle,  the 
sum  of  the  three  angles  is  equal  to  4  right  angles; 
and  this  sum,  minus  two  riglit-  angles,  is  represented 
by  4  — 2,  or  2  ;  therefore,  the  surface  of  the  triangle  is 
equal  to  tw^o  tri-rectangular  triangles,  or  to  the  fourth 
part  of  the  surface  of  the  entire  sphere. 

jScholium  2.  The  same  proportion  which  exists  between 
the  spherical  triangle  ABC,  and  the  tri-rectangular  triangle, 
exists  also  between  the  spherical  pyramid  which  has  ABO 
for  its  base,  and  the  tri-rectangular  pyramid.  The  triedral 
angle  of  the  pyramid  is  to  the  triedral  angle  of  the  tri- 
rcc':angular  pyramid,  as  the  triangle  ABC  to  the  tri-rectan- 
gidar  triangle.  From  these  relations,  the  following  conse- 
quences are  deduced. 

First.  Two  triangular  spherical  pyramids  are  to  each 
other  as  their  bases :  and  since  a  polj^gonal  pyramid  may 
always  be  divided  into  a  certain  number  of  ti'iangnlar 
pyiamids,  it  follows  that  any  two  spherical  pj'ramids  are  to 
each  other,  as  the  polygons  which  form  their  bases. 


2U  GEOMETRY. 

Second  The  polyedral  augles  at  the  vertices  of  these 
pyramids,  aie  also  as  their  bases ;  hence,  for  comparing 
any  two  poh'edral  angles,  we  have  merely  to  place  their 
vertices  at  the  centres  of  two  equal  spheres;  the  angles  are 
to  each  other  as  the  spherical  j^olygous  intercepted  between 
their  faces. 

The  vertical  angle  of  the  tri-rectangular  pyramid  is 
formed  by  three  planes  at  right  angles  to  each  other:  this 
angle,  which  may  be  called  a  rigJit  p^Ayedral  anr/le,  Avili 
serve  as  a  very  natural  unit  of  measure  for  all  other  poly- 
edral angles.  If,  for  example,  the  area  of  the  triangle  is 
I  of  the  tri-rectangular  triangle,  the  corresponding  trie 
dral  angle  is  also  J  of  the  right  polyedral  angle. 

rEorosrnox  xix.    TiiEOEEii. 

The  surface  of  a  spherical  polijrjon  is  equal  to  the  excess  of  the 
sura  of  all  its  anrjles^  over  two  ri>jld  angles  taken  as  many 
times  as  there  are  sides  in  the  polygon  less  twOj  multiplied 
hy  iJie  tri-rectangular  triangle. 

Let  ABODE  be  a  spherical  polygon. 

From  one  of  the  vertices  ^4,  let- 
diagonals  AC,  AD^  be  drawn  to  the 
other  vertices ;  the  polygon  ABODE 
will  be  divided  into  as  many  tri- 
angles less  two,  as  it  has  sides. 

Now,  the  surface  of  each  triangle 
is  equal  to  the  sum  of  all  its  angles  less  Iavo  right  angles, 
into  the  tri-rectangular  triangle.  The  sum  of  the  angles 
of  all  the  triangles  is  the  same  as  that  of  all  the  angles  of 
the  iX)lygon;  hence,  the  surface  of  the  polj-gon  is  equal  to 
the  sum  of  all  its  angles,  diminished  by  twice  as  many 
right  angles  as  it  has  sides  less  two,  into  the  tri-rectangu- 
lar triangle. 

Scholium.  Let  s  be  the  sum  of  all  the  angles  of  a  spheri 
cal  polygon,  ?i  the  number  of  its  sides,  and  T  the  tri-rect- 
ungular  triangle  ;  the  right  angle  being  taken  as  unity,  the 
surface  of  the  polygon  will  be  equal  to 

(5-2  {n-2;)\xT={s-2n-{-4)xT. 


APPENDIX. 


NOTE    A.  — Page  22. 


\ 


A  Demonstration  is  a  train  of  logical  arguments 
brouglit  to  a  conclusion.  The  bases  or  premises  of  a 
demonstration,  are  definitions,  axioms,  propositions  pre- 
viously established,  and  hypotheses.  The  arguments  are 
the  links  which  connect  the  premises,  logically,  with  the 
conclusion  or  ultimate  truth  to  be  proved. 

In  Geometry  we  employ  two  kinds  of  demonstration — 
the  Direct,  and  the  Indirect  or  the  method  involving  the 
Heductio  ad  absurd  urn. 

These  are  also  called  Positive  and  Negative  Demonstra- 
tions. In  the  direct  method,  the  premises  are  definitions, 
axioms,  and  previous  propositions ;  and  by  a  pnjcess  of 
logical  argumentation,  the  magnitudes  of  which  something 
is  to  be  proved,  are  shown  to  bear  the  mark  by  which 
that  may  always  be  inferred,  or,  in  other  words,  are  shown 
to  fall  under  some  definition,  axiom,  or  proposition,  pre- 
viously laid  down.  The  direct  demonstration  may  be 
divided  into  two  classes: 

1st.  Where  the  argument  depends  on  suj)erposition — • 
that  is,  on  the  coincidence  of  magnitudes  when  applied  the 
one  to  the  other:    and 

2dly.  Where  it  depends  on  addition  and  subtraction, 
or  immediately  on  principles  previous!}^  laid  down. 

The  indirect  method  rests  on  a  hypothesis.  1'his  hyjio- 
thesis  is  combined  in  a  process  of  logical  argumentation,- 
with  definitions,  axioms,  and  previous  propositions,  until 
a  conclusion  is  obtained,  which  agrees  or  disagrees  with 
seme  known  truth.  Now,  if  the  conclusion  so  deduced,  is 
excluded  from   the    truths  previously  established,   that  is,  if 


246  APPENDIX. 


\ 


it  is  opposed  to  any  of  them,  then  it  follows  that  tlie  hy- 
pothesis, leading  to  a  result  contradictory  to  such  truth, 
must  be  false.  In  the  indirect  demonstration,  therefore,  the 
conclusion-  is  compared  Avith  the  truths  known  antecedently 
to  the  proposition  in  question  ;  if  it  disagrees  with  any  of 
them,  the  hypothesis  is  false. 

We  have  examples  of  the  first  class  of  the  direct  demon- 
stration in  the  reasoning  which  establishes  Pi'opositions  V. 
and  VI. — and  of  the  second  class  in  that  which  establishes 
Propositions  I,  and  IV.  AVe  have  also  examples  of  the 
indirect  method  in  the  demonstrations  of  Propositions  II. 
and  III. 

It  is  often  supposed,  though  erroneously,  that  the  indi- 
rect demonstration  is  less  conclusive  and  satisfactory  than 
the  direct.  This  impression  is  simply  the  result  of  a  want 
of  })roper  analysis.  For  example :  in  the  demonstration 
of  Proposition  II.  we  propose  to  prove  "  that  two  sti-aight 
lines  having  two  points  in  common  coincide  throughout 
their  whole  extent."  Now,  it  is  evident  that  they  either 
coincide  or  separate.  If  they  separate,  they  must  separate 
at  some  point,  as  C.  But  the  snpposUion  or  Jtupothcsis  of 
their  separating  at  this  ]>oint,  involves  the  conclusion,  that 
a  7)aW  is  equal  to  the  icliole,  which  is  conti'ary  to  Axiom  8, 
and  therefore  untrue:  Hence,  they  do  not  separate,  and 
Hierefore^  they  coincide.  Similar  remarks  apply  to  all  indi- 
rect demonstrations. 

In  both  kinds  of  demonstrations  the  premises  and  ccn- 
clusion  agree:  that  is,  they  are  both  true  or  both  false, 
the  reasoning  or  argument  in  both  being  supposed  strictly 
logical. 

For  a  more  full  discussion  of  this  subject,  see  Davisa* 
Logic  of  Mathematics. 


APPENDIX.  247 


THE    REGULAR    POLYEDRONS. 

A  Eegulak  Polyedrox  is  one  whose  faces  are  all 
equal  regular  polygons,  and  whose  polyedral  angles  are  all 
equal  to  each  other. 

1.  The  Tetraedrox,  or  regular  pyramid^  is  a  solid 
bounded  by  four  equal  equilateral  triangles. 

2.  The  Hexaedro:^,  or  Cube,  is  a  solid  bounded  by 
six  equal  squares. 

3.  The  OcTAEDRo:^,  is  a  solid  bounded  by  eight  equal 
equilateral  triangles. 

4.  The  DoDECAEDRON,  is  a  solid  bounded  by  twelve 
equal  and  regular  pentagons. 

5.  The  IcosAEDRON  is  a  solid  bounded  by  twenty- 
equal  equilateral  triangles. 

First.  If  the  faces  are  equilateral  triangles,  polj'edrons 
may  be  constructed  bounded  by  such  triangles  and  will 
have  polj'cdral  angles  contained  either  by  three,  four  or 
five  of  them :  hence  arise  three  regular  polycdral  bodies, 
viz:  the  tetraedron^  the  octaedron^  and  the  icosaedroii^  and  no 
others  can  be  constructed  with  equilateral  ti'iangles.  For, 
each  angle  of  an  equilateral  triangle  being  equal  to  a  third 
part  of  two  right,  six  such  angles  about  the  vertex  of  a 
polyedral  angle  would  be  equal  to  four  riglit  angles,  which 
is  impossible  (b.  vi.,  P.  20, 

Secondly.  If  the  faces  are  squares,  their  angles  may  be 
arranged  by  threes :  hence,  results  the  hexaedron^  or  cube. 
Four  angles  of  a  square  are  equal  to  four  right  angles, 
and  cannot  form  a  polyedral  angle. 

Thirdly.  In  fine,  if  the  faces  are  regular  pentagons, 
their  angles  likewise  may  be  arranged  by  threes :  the 
regular  dodecaedron  will  result. 

We  can  proceed  no  farther :  three  angles  of  a  regular 
hexagon  are  equal  to  four  right  angles ;  three  of  a  hepta- 
gon are  greater. 


218 


APPEXDIX. 


Tlcncc,  there  can  only  be  five  rcgj/ar  polyetlrons ;  tbreo 
forme*,!  with  equilateral  triangles,  c-ijo  witli  squares,  and 
one  with  pentagons. 


CONSTEUCTIOX   OF   THE   TETEAEDROX. 

Let  ABC  be  the  equilateral  triangle  which  is  to  form 
one  face  of  the  tetraeclron.  At  the  point  0,  the  centre  of 
this  triangle,  erect  OS  perpendicular  to  the  plane  ABC) 
terminate  this  perpendicular  in  S,  so  that  AS~AB\  draw 
SB^  SC ;    the'p3'ramid  S-ABC  is  the  tetraedron  required. 

For,  by  reason  of  the  equal  distan- 
ces OA,  OB,  OC,  the  oblique  lines  SA, 
SB,  SC,  cut  off  equal  distances  esti- 
mated from  the  foot  of  the  perpendic- 
ular SO,  and  consequently  are  equal 
(b.  VI.,  P.  5).  One  of  them  SA=AB\ 
hence,  the  four  faces  of  the  pyramid 
S-ABC,    are    triangles,    equal    to    the 

given  triangle  ABC.  The  triedral  angles  of  this  pyramid 
are  all  equal,  because  each  of  thein  is  bounded  by  three 
equal  plane  angles  (b.  vi.,  r.  21,  s.  2);  hence,  this  pyramid 
is  a  regular  tetraedron. 


COXSTRUCTIOX   OF   THE   IIEXAEDRON. 


\ 

N 

^C 

\] 

\ 

B 


Let  ABCD  be  a  given  square.  On  the 
base  ABCD,  construct  a  right  prism  Avhose 
altitude  AE  shall  be  equal  to  the  side 
AB.  The  faces  of  this  prism  will  evident- 
ly be  equal  squares ;  and  its  triedral  an- 
gles all  equal,  each  being  formed  with 
three  equal  faces :  hence,  this  prism  is  a 
regular  hexaedron  or  cube. 

The  following  propositions  can  be  easily  proved. 

1.  Any  regular  polyedron  may  be  divided  into  as  many 
right  pyramids  as  the  pol^'edron  has  faces  ;  the  common 
vertex  of  these  pyramids  will  be  the  centre  of  the  polye- 


APPLICATION    OF    ALGEBRA.  249 

dron;    and  at  the  same  time,  that  of  an  inscribed  and  of 
a  circumscribed  sphere. 

2.  The  solidity  of  a  regular  polyedron  is  equal  to  its 
surface  multiplied  by  a  third  part  of  the  radius  of  the 
inscribed  sphere. 

3.  Tvro  regular  polyedrons  of  the  same  name,  are  two 
snnilar  solids,  and  their  homologous  dimensions  are  pro- 
portional ;  hence,  the  radii  of  the  inscribed  or  the  cii-cuin- 
scribed  spheres  are  to  each  other  as  the  edges  of  the  poly- 
edrons. 

4.  If  a  regular  polyedron  be  inscribed  in  a  sphere,  the 
planes  di-awn  from  the  centre,  through  the  diiVcrcnt  edges, 
will  divide  the  surface  of  the  S})here  into  as  many  spheri- 
cal polygons,  all  equal  and  similar,  as  the  polyedron  has 
faces. 


APPLICATION   OF   ALGEBRA 

TO    TUE 

SOLUTION  OF  GEOMETRICAL  PROBLEMS. 

A  Proble:m  is  a  question  which  requires  a  solution. 
A  geometrical  problem  is  one,  in  which  certain  parts  of  a 
geometrical  figure  are  given  or  known,  from  which  it  is 
required  to  determine  certain  other  parts. 

When  it  is  proposed  to  solve  a  geometrical  problem  by 
means  of  Algebra,  the  given  parts  are  represented  b}-  the 
first  letters  of  the  alphabet,  and  the  required  parts  by  the 
final  letters.  The  geometrical  relations  v.dncli  subsist  be- 
tween the  known  and  required  parts  furnish  the  equations 
of  the  problem.  The  solution  of  these  equations,  when  so 
formed,  gives  the  solution  of  the  problem. 

No  general  rule  can  be  given  for  forming  the  equations. 
The  equations  must  be  independent  of  each  other,  and 
their  number  equal  to  that  of  the  unknown  quantities  in- 
troduced (Alg.,  Art.  lOo).  Experience,  and  a  careful  exami- 
nation of  all  the  conditions,  whether  explicit  or  implicit 
(Alg.,  Art  94),  will  serve  as  guides  in  stating  the  questions, 
to  which  may  be  added  the  following  general  directions. 


250 


APPENDIX. 


1st.  Draw  a  figure  winch  sliall  represent  all  the  given 
parts,  and  all  the  required  parts.  Then  draw  such  other 
lines  as  will  enable  us  to  establish  t\\e  necessary  relations 
betvv'eeii  them.  If  an  angle  is  given,  it  is  generally  best  to 
let  fall  a  perpendicular  that  shall  lie  opposite  to  it ;  and 
this  perpendicular,  if  possible,  should  be  drawn  from  the 
extremity  of  a  given  side. 

2d.  AVhen  two  lines  or  quantities  are  connected  in  the 
same  way  with  other  parts  of  the  figure  or  problem,  it  is 
in  general,  not  best  to  use  either  of  them  separately  ;  but 
to  use  their  sum,  their  difference,  their  product,  their  quo- 
tient, or'  perhaps  another  line  of  the  figure  with  which 
they  are  alike  connected. 

3d.  When  the  area,  or  perimeter  of  a  figure,  is  given, 
it  is  sometimes  best  to  assun^e  another  figure  similar  to  that 
proposed,  having  one  of  its  sides  equal  to  unity,  or  some 
other  known  quantity.  A  comparison  of  the  two  figures 
will  often  give  a  required  part.  We  will  add  the  follow- 
ing problems." 

PROBLEM  I. 


In  a  rirjhl-avrjled  triaDgle  BA  C,  having  given  the  base  BA^ 
and  the  sum  of  the  hyjiotlienuse  and  perpendicular,  it  is 
required  to  find  the  hypoOienuse  and  perpendicular. 

Put  BA  =  c  =  3,  BC  =  x^  AC  =  y,  and  the  sum  of  the 

bypothenuse  and  perpendicular  equal  to  s  =  9. 

Then,  cc  +  ?/  =  5  =  9, 

and  (b.  IV.,  P.  11),  x-  =  f  +  c-. 

From  1st  equ  :  x  =  s  —  y, 

and  ar  =  s-  —  2sf/  +  ^-. 

By  subtracting,  0  =  s-  —  2sy  —  c-, 

or,  'Zsy  =  s' 

s"-  -^  cr 
hence,  y  =  — ^ =  4:  =  AC. 

Therefore. 


Zsy  =  s-  —  c ; 


2s 


a:  +  4  =  9,  or  .T  --  5  =  BC. 


*  The  followii.ff  problems  are  selected  from  ITutton's  Application  of  Algebra  to 
Geometry;  and  the  examples  iu  Mensuration,  from  his  treatise  or)  that  aubject. 


APPLICATION    OF    ALGEBRA, 


251 


TKOBLEM  II. 

In  a  rvjld'anrjled  triangle^  having  given  the  hypotliennse^  and 
the  sum  of  the  hase  and  ^ye;'^>ie;<c/<c«/ar,  to  find  tliese  two 
sides. 


Put   BC  —  a  =  5,    BA  =  cc,  AC  —  y^    and   tlie   surn   of 
the  base  and  perpendicular  —  s  =  1, 

Then,  a:  +  ?/  =  5  =  7, 

and  it-  +  if  =  a-. 

From  first  equation,         x  =  s  —  y, 

or,  X-  =  6-  -  25^  +  7/ ; 

Hence,  f  =  cC-  -  s^  +  2sy  -  f-^ 

or,  2/  -  2.sy  =  iC-  -  s" ; 

(J?  -  .s^ 


or, 


y  -  sy 


By  completing  the  square  y'^  —  sy  +  \ir  =  Ja-  —{s^ 


or. 


Hence, 


y  =  Is  ±:  V^(r  —  Is-  =  4  or  3. 
^  =  -2-5  ^  V^a-  -  is-  =  3  or  4. 


PROBLEM   III. 

■Z/i  a  rectangle,  having  given  the  diagonal  arid  2^crinieter^  to  find 

the  sides. 


Let  ABCD  be  the  proposed  rectangle. 
Put  A  C  =  d  =  10,  the  perimeter  =2a  =  28, 
or  AB  +BC  =  a  =  14  :    also  put  AB  =  x, 
and  BC  =  ?/. 

Then,  a:^  +  ?/-  =  c?2, 

and  cc  +  ?/  =  a. 

From  which  equations  we  obtain, 


and 


y  =  \a  ±  ^fyr^  -  |.,2  =  8  or  6, 
X  =  Jazp  Vic/-!  -  1^^  =  G  or  8. 


252 


APPEXDIX 


TROBLEM   IV 


Havinrj  given  the  base  and  i^f^rpendicular  of  a  triangle^  to  find 
iJie  side  of  an  inscribed  square. 

Q 

Let  ABC  be  the  triangle,  and 
IIEFG  the  inscribed  square.  Put 
AB  =b,  CD  =  «,  and  HE  or  GH 
=  X  :    then   CI  =  a  —  x. 

We  have  by  siniihir  triangles 

AB    :     CD    w     GF 
or,  b     :     a     :  :     X 

Hence,  cJj  —  bx  =  ax, 

or,  X  =  — - — r  =  the  side  of  the  inscribed  square: 

a  +  0  ^ 

which,  therefore,  depends  onl}'  on  the  base  and  altitude  of 
the  triangle. 


CI 


a  —  X 


rFwOBLE^I   V. 


fn  an  equ.ilateral  triangle,  having  given  the  lengths  of  the  three 
perpendiculars  drawn  from  a  point  tvithin,  on  thte  tlirec 
sides:   to  determine  the  sides  of  the  triangle. 

Let  ABC  be  an  equilateral  trian- 
gle :  DG,  DE  and  DF  the  given  per 
pendiculars  let  fall  from  D  on  the 
sides.  Draw  DA,  DB,  DC,  to  the 
vertices  of  the  angles,  and  let  fall  the 
perpendicular  C II  on  the  base.  Let 
DG  =  a,  DE  =  b,  and  DF  =  c  :  put 
one  of  the  equal  sides  AB  ~  2.r  ;  hence,  All  =  x,  and 
CH  =  VAC'  -^fP  =  V^^i?  =  V^  =  X  Vs. 

Now,  since  the  area  of  a  triangle    is    equal    to   half  its 
base  into  the  altitude,  (b.  iv.,  p.  6), 

\ABx  CII  =  X  X  X  \^=  a^  VT=  triangle  ACB, 
lAB  X  DG  =  X  X  a  =  ax  =^  triangle  ADB^ 
\BC  X  DE=  X  Xb  ^  bx  =  triangle  BCD, 

{AC  X  DF  —  X  X  c  =  ex  =  triangle' ^i(7i) 


APPLICATION    OF    ALGEBRA.  253 

But   the   last   three   triangles   make   up,  and  are   conse- 
qaently  equal  to,  the  first ; 

hence,       or  Vz  =^  ax  -\-  l>x  ■\-  ex  =  x  (a  -^r  h  -^  c)  \ 
or,  X  V^=  a  -i-  b  +  c: 

a  -\-  h  -i-  c 


therefore,  x  = 


V3 


Remark.  Since  the  perpendicular  CII  is  equal  to  xVs, 
it  is  consequently  equal  a  -{-  b  -\-  c  :  that  is,  the  perpendic- 
ular let  fall  from  either  angle  of  an  equilateral  triangle  on 
the  opposite  side,  is  equal  to  the  sum  of  the  three  perpen- 
diculars let  fall  from  any  point  "within  the  triangle  on  the 
sides  respectively. 

Problem  YI. — In  a  right-angled  triangle,  having  given 
the  base  and  the  difference  between  the  hypothenuse  and 
perpendicular,   to  lind  the  sides. 

pROJJLEM  YII. — In  a  riii'lit-auG^led  trian2;le,  havins]^  mven 
the  hypothenuse,  and  the  diil'erence  between  the  base  and 
perpendicular,  to  determine  the  triangle. 

Problem  YIII. — Having  given  the  area  of  a  rectangle 
inscribed  in  a  given  triangle ;  to  determine  the  sides  of 
the   rectangle.        ' 

Problem  IX. — In  a  triangle,  having  given  the  ratio  of 
the  two  sides,  to^'ether  with  both  the  segments  of  the  base 

JO  o 

made  by  a  perpendicular    from    the   vertical  angle ;    to  de- 
termine the  triangle. 

Prohlem  X. — In  a  triangle,  having  given  the  base, ,  the 
sum  of  the  two  other  sides,  and  the  length  of  a  line 
drawn  from  the  vertical  angle  to  the  middle  of  the  base; 
tc  hud  the  sides  of  the  triangle. 

Pro]5LEM  XT. — In  a  triangle,  having  given  the  two 
sides  about  the  vertical  angle,  together  with  the  line  bisect- 
ing that  angle  and  terminating  in  the  ba.^f  •  to  lind  tho 
base. 

Problem  XII. — To  determine  a  riiiht-anirled  trianjile, 
having  given  the  lengths  of  two  lines  drawn  from  the 
acuto  angles  to  the  middle  of  the  opposite  sides. 


251  APPEXDIX. 

Problem  XIIL — To  determine  a  right-angled  triangle, 
having  given  the  perimeter  and  the  radios  of  the  inscribed 
circle. 

Pkoble^[  XIT. — To  determine  a  triangle,  having  given 
the  buse,  the  perpendicular,  and  the  ratio  of  the  two  sides. 

Problem  XV. — To  determine  a  right-angled  triangle, 
having  given  the  hvpothenuse,  and  the  side  of  the  inscribed 
square. 

PROBLE>r  XVI. — To  determine  the  radii  of  three  equal 
circles,  described  within  and  tangent  to,  a  given  circle,  and 
also   tangent   to  each  other. 

Problem!  XVII. — In  a  right-angle  triangle,  having  given 
the  perimeter  and  the  perpendicular  let  fall  from  the  right 
angle  on  the  h3'pothenuse,  to  determine  the  triangle. 

Problem  XVIII. — To  determine  a  right-angled  triangle, 
having  given  the  hvpothenuse  and  the  difference  of  two 
lines  drawn  from  the  two  acute  angles  to  the  centre  of  the 
inscribed  circle. 

Problem  XIX. — To  determine  a  triansrle,  havino:  criven 
the  base,  the  perpendicular,  and  the  difference  of  the  two 
other  sides. 

Problem  XX. — To  determine  a  triangle,  having  given 
the  base,  the  perpendicular,  and  the  rectangle  of  the  two 
sides. 

PROBLE:^r  XXL — To  determine  a  friangle,  having  given 
the  lengths  of  three  lines  drawn  from  the  three  angles  to 
the  middle  of  the  opposite  sides. 

Proble:n[  XXn. — In  a  triangle,  having  given  the  three 
sides,  to  find  the  radius  of  the  inscribed  circle. 

Problem  XXIII.— To  determine  a  right-angled  triangle, 
having  given  the  side  of  the  inscribed  square,  and  the 
radius  of  the  inscribed  circle. 

Problem  XXIV. — To  determine  a  right-angled  triangle, 
iaving  given  the  hvpothenuse  and  radius  of  the  inscribed 
circle. 

Problem  XXV. — To  determine  a  triangle,  havmg  given 
the  base,  the  line  bisecting  the  vertical  angle,  and  the  diam 
eter  of  the  circumscribing  circle. 


•4 


PLANE  TRIGONOMETRY. 


INTKODUCTIOX 


OF    LOGARITHMS. 

1.  The  logarithm  of  a  ^lumher  is  the  exponent  of  the  poiucT 
to  which  it  is  necessary  to  raise  a  fixed  nuraher^  in  order  to 
produce  the  first  number. 

This  fixed  number  is  called  the  hase  of  the  system,  and 
may  be  any  number  except  1 :  in  the  common  system,  10 
is  assumed  as  the  base. 

2.  If  we  form  those  powers  of  10,  which  are  denoted 
by  entire  exponents,  we  shall  have 

10"=  1       10'  =  10  ,        10' =  1000 

10' =  100,        lO' =  10000,  &c.,  &c., 

From  the  above  table,  it  is  plain,  that  0,  1,  2,  3,  4,  (5:c., 
are  respectively  the  logarithms  of  1,  10,  100,  1000,  10000, 
&c. ;    we   also   see,  that  ihe   logarithm   of  any  number   be- 
tween 1  and  10,  is  greater  than  0  and  less  than  1 :    thus, 
log  2  =  0.301030. 

The  logarithm  of  any  number  greater  than  10,  and  less 
than  100,  is  greater  than  1  and  less  than  2  :    thus, 
log  50  =  1.698970. 

The  logarithm  of  any  number  greater  than  100,  and 
less  than  1000,  is  greater  than  2  and  less  than  3 :   thus, 

log  126  =  2.100371,  &;c. 


250  PLANE    TRIGOXOMETEY. 

If  tLe  above  principles  be  extended  to  other  numbers, 
it  Avill  a])pt'ar,  that  the  logarithm  of  any  number,  not  an 
exact  power  of  ten,  is  made  up  of  two  parts,  an  entire  and 
a  deciiiad  pari.  The  entire  'part  is  called  the  diaracteristic 
cf  tie  lofjarit/un,  and  is  always  one  less  than  the-nmnber  of 
places  oi'  ligares  in  the  given  number. 

3.  The  principal  use  of  logarithms,  is  to  abridge  nu- 
merical computations. 

Let  M  denote  any  number,  and  let  its  logarithm  be 
denoted  by  rn ;  also  let  xV  denote  a  second  number  ^vhose 
logarithm  is  n  ;   then,  from  the  definition,  -we  shall  have, 

10'"  =  J/     (1)  10"  =  xY     (2). 

Multiplying  equations  (1)  and  (2),  member  by  member, 
we  have, 

10"'+"  =  J/x  X  or,  m  +  ?2=log  (J/X  X) ;     hence, 

Tlie  snm  of  the  logaritlims  of  any  tico  numhers  is  equal  to 
the  lorjariOun  of  their  product. 

4.  Dividing  equation  (1)  by  equation  (2),  member  by 
member,  we  have, 

10        =-rror,  m  — n  =  log-y:    hence, 

The  logarithm  of  lite  quotient  of  tKO  numhers,  is  equal  to 
the  logarithm  of  the  dividend  diminished  hy  the  logarithm  of 
the  divisor. 

5.  Since  the  logarithm  of  10  is  1,  the  logarithm  of  the 
product  of  any  number  by  10,  icill  he  greater  hy  1  tlian  the 
logarithm  of  t/ait  number ;  also,  tlie  logarithm  of  the  quotient 
of  any  number  divided  hy  10,  will  he  less  hy  1  than  tlie 
logarithm  of  that  number. 

Similarly,  it  may  be  sho-wm  that  if  any  number  be  mul- 
tiplied by  one  hundred,  the  logarithm  of  the  product  will 
be  greater  by  2  than  the  logarithm  of  that  number ;  and 
if  any  number  be  divided  by  one  hundred,  the  logarithm 
of  the  quotient  will  be  less  by  2  than  the  logarithm  of 
that  number,  and  so  on. 


INTRODUCTION.  257 


EXA 

lMPLEI 

3. 

log  327 

is 

2.514548 

log  32.7 

u 

1.514548 

log  3.27 

(( 

6.514548 

log  .327 

(( 

1.514548 

W  .0327 

u 

2.514548 

From  the  above  examples,  we  see,  that  in  a  number 
composed  of  an  entire  and  decimal  part,  we  may  change 
the  place  of  the  decim.ai  point  without  changing  the  deci- 
mal part  of  the  logarithm;  but  the  characteristic  is  dimirtr 
ished  hy  1  for  every  i:)lace  that  the  decimal  point  is  removed  to 
Hie  left. 

In  the  lo2rarithm  of  a  decimal,  the  characteristic  becomes 
negative,  and  is  numerically  1  greater  than  the  number  of 
ciphers  immediately  after  the  decimal  point.  The  negative 
sign  extends  only  to  the  characteristic,  and  is  written  over 
it,  as  in  the  examples  given  above. 

TABLE   OF  LOGARITHMS. 

6.  A  table  of  logarithms,  is  a  table  in  which  are  writ- 
ten the  logarithms  of  all  numbers  between  1  and  some 
given  number.  The  logarithms  of  all  numbers  between  1 
and  10,000  are  given  in  the  annexed  table.  Since  rules 
have  been  given  for  determining  the  characteristics  of 
logarithms  by  simple  inspection,,  it  has  not  been  deemed 
necessary  to  write  them  in  the  table,  the  decimal  part 
only  being  given.  The  characteristic,  however,  is  given 
for  all  numbers  less  than  100. 

The  left  hand  column  of  each  page  of  the  table,  is  the 
column  of  numbers,  and  is  designated  by  the  letter  N; 
the  logarithms  of  these  numbers  are  placed  opposite  them 
on  the  same  horizontal  line.  The  last  column  on  each 
page,  headed  D,  shows  the  difference  between  the  legit- 
rithms  of  two  consecutive  numbers.  This  difference  is 
found  by  subtracting  the  logarithm  under  the  column 
headed  4,  from  the  one  in  the  column  headed  5  in  tht* 
same  horizontal  line,  and  is  nearly  a  mean  of  the  diiler- 
euces  of  any  two  consecutive  logarithms  on  this  line, 

17 


258  PLANE    TRIGONOMETK Y. 

To  find,  from  the  table,  the  logarithm  of  any  number. 

7.  If  the  number  is  less  than  100,  look  on  the  first  })age 
of  tlie  table,  in  the  column  of  numbers  under  N,  until  tho 
number   is   found :    the   number   opposite   is   the  logarithm 
ought :    Thus. 

log  9  =  0.95^243. 


When  the  number  is  greater  than  100   and  less  than  10000. 

8.  Find  in  the  column  of  numbers,  the  first  three  figures 
of  the  given  number.  Then  pass  across  the  page  along  a 
horizontal  line  until  you  come  into  the  column  under  the 
fourth  figure  of  the  given  number :  at  this  place,  there  are 
four  figures  of  the  required  logarithm,  to  which,  two  figures 
taken  from  the  column  marked  0,  are  to  be  prefixed. 

If  the  four  figures  already  found  stand  opposite  a  row 
of  six  figures  in  the  column  marked  0,  the  two  left  hand 
figures  of  the  six,  are  the  two  to  be  prefixed ;  but  if  they 
stand  opposite  a  row  of  only  four  figures,  you  ascend  the 
column  till  you  find  a  row  of  six  figures ;  the  two  left 
hand  figures  of  this  row  are  the  two  to  be  prefixed.  K 
you  p.refix  to  the  decimal  part  thus  found,  the  character- 
istic, you  vnll  have  the  logarithm  sought:    Thus, 

log     8979  =  3.953228 
log  .08979  =  2.953228 

If,  however,  in  passing  back  from  the  four  figures  found, 
t*o  the  0  column,  any  dots  be  met  with,  the  two  figures 
to  be  prefixed  must  be  taken  from  the  horizontal  line  di- 
rectly below :    Thus, 

log   3098  =  3.491081 
log  30.98  =  1.491081 

If  the  logarithm  falls  at  a  place  where  the  dots  occur, 
0  must  be  written  for  each  dot,  and  the  two  figures  to  be 
prefixed  are,  as  before,  taken  from  the  line  below :    Thus, 

log   2188  =  3.340047 
log   2188  =  1.340047 


INTRODUCTION.  259 

When  the  number  exceeds  10,000. 

9.  The  characteristic  is  determined  by  the  rules  ah*eady 
given.  To  find  the  decimal  part  of  the  logarithm :  place 
a  decimal  point  after  the  fourth  figure  from  the  left 
hand,  converting  the  given  number  into  a  whole  number 
and  decimal.  Find  the  logarithm  of  the  entire  part  b}'  the 
rule  just  given,  then  take  from  the  right  hand  column  of 
the  page,  under  D,  the  number  on  the  same  horizontal 
line  with  the  logarithm,  and  multiply  it  by  the  decimal 
part ;  add  the  product  thus  obtained  to  the  logarithm  al- 
ready found,  and  the  sum  will  be  the  logarithm  sought. 

If,  in  multiplying  the  number  taken  from  the  column 
D,  the  decimal  part  of  the  product  exceeds  .5,  let  1  be 
added  to  the  entire  part;  if  it  is  less  than  .5,  the  decimal 
part  of  the  product  is  neglected. 

EXAMPLE. 

1.   To  find  the  logarithm  of  the  number  672887. 

The  characteristic  is  5.;  placing  a  decimal  point  after 
the  fourth  figure  from  the  left,  we  have  6728.87.  The 
decimal  part  of  the  log  6728  is  .827886,  and  the  corres- 
ponding number  in  the  column  D  is  65;  then  65X.87= 
56.55,  and  since  the  decimal  part  exceeds  .5,  we  have  57 
to  be  added  to  .827886,  which  gives  .827943. 

Hence,  log     672887  =  6.827943 

Similarly,  log  .0672887  =  2.827943 

The  last  rule  has  been  deduced  under  the  supposition 
that  the  difference  of  the  numbers  is  proportional  to  the 
difterence  of  their  logarithms,  which  is  sufl&ciently  exact 
within  the  narrow  limits  considered. 

In  the  above  example,  Qb  is  the  difference  between  the 
logarithm  of  672900  and  the  logarithm  of  672800,  that  is, 
it  is  the  difference  between  the  logarithms  of  two  numbers 
which  differ  by  100. 

We  have  then  the  proportion 

100    :     87    ::     65    :     56.55, 

hence,  56.55  is  the   number  to  be  added  to  the  logarithm 
before  found. 


260  PLANE    TRIGONOMETRY. 

To  find  from  the  table  the  number  corresponding  to  a  given 
logarithm. 

10.  Search  in  the  columns  of  logarithms  for  the  decimal 
part  of  the  given  logarithm :  if  it  cannot  be  found  in  the 
table,  take  out  the  number  corresponding  to  the  next  less 
logarithm  and  set  it  aside.  Subtract  this  less  logarithm 
from  the  given  logarithm,  and  annex  to  the  remainder  as 
many  zeros  as  may  be  necessary,  and  divide  this  restJt  by 
the  corresponding  number  taken  from  the  column  marked 
D,  continuing  the  division  as  long  as  desirable :  annex  the 
quotient  to  the  number  set  aside.  Point  off,  from  the  left 
hand,  as  many  integer  figures  as  there  are  units  in  the 
characteristic  of  the  given  logarithm  increased  by  1 ;  the 
result  is  the  requii-ed  number. 

If  the  characteristic  is  negative,  the  number  will  be 
entirely  decimal,  and  the  number  of  zeros  to  be  placed  at 
the  left  of  the  number  found  from  the  table,  will  be  equal  to 
the  number  of  units  in  the  characteristic  diminished  by  1. 

This  rule,  like  its  converse,  is  founded  on  the  supposi- 
tion that  the  difference  of  the  logarithms  is  proportional 
to  the  difference  of  their  numbers  within  narrow  limits. 

EXAMPLE. 

1.  Find  the  number  corresponding  to  the  logarithm 
S.233568. 

The  decimal  part  of  the  given  logarithm  is      .233568 
The  next  less  logarithm  of  the  table  is  .233501:, 

and  its  corresponding  number  1712.  

Their  difference  is  -  •  -  -  64 

Tabular  difference  253)6400000(25 

Hence,  the  number  sought  1712.25. 

The   number   corresponding   to   the   logarithm  3.233568 

is  .00171225. 

2.  What  is  the  number  corresponding  to  the  logarithm 
2.785-107?  Ans.  .06101084. 

3.  What  is  the  number  corresponding  to  the  logarithm 
1.846741?  Ans.  .702658. 


INTRODUCTION.  26J 

MULTIPLICATION   liY   LOGARITHMS. 

11.  When  it  is  required  to  multiply  numbers  by  meana 
of  their  logarithms,  we  first  find  from  the  table  the  loga- 
rithms of  the  numbers  to  be  multiplied ;  we  next  add 
these  logarithms  together,  and  their  sum  is  the  logarithm 
of  the  product  of  the  numbers  (Art.  3). 

The  term  sum  is  to  be  understood  in  its  algebraic 
sense;  therefore,  if  any  of  the  logarithms  have  negative 
characteristics,  the  dift'erence  between  their  sum  and  that 
of  the  positive  characteristics,  is  to  be  taken ;  the  sign  of 
the  remainder  is  that  of  the  greater  sum. 

EXAMPLES. 

1.   Multiply  23.14  by  5.062. 

log  23.14  =  1.364363 
log  5.062  =  0.704322 

Product,  117.1347  .  .  .  27o^S685 


2.  Multiply  3.902,  597.16,  and  0.0314728  together. 

log  3.902  =  0.591287 

log        697.16  =  2.776091 
log  0.0314728  =  2.497936 

Product,  73.3354  ....  1.865314 

Here,  the   2   cancels  the  +  2,  and   the  1  carried   from 
the  decimal  part  is  set  down. 

3.  Multiply  3.586,  2.1046,  0.8372,  and  0.0294  together. 

log  3.586  =  0.554610 
log  2.1046  =  0.323170 
log  0.8372  =  1.922829 
log  0.0294  =  2.468347 


Product,  0.1857615  .  .  1.268956 


In  this  example  the  2,  carried  from   the  decimal   part, 
cancels  2,  and  there  remains  1  to  be  set  down. 


262  PLANE    TIUGOXOMETEY. 


DIVISION   OF   NUMBERS    BY   LOGARITHMS. 

12.  When  it  is  required  to  divide  numbers  by  means 
of  their  logarithms,  we  have  only  to  recollect,  that  the 
subtraction  of  logarithms  corresponds  to  the  division  of 
their  numbers  (Art.  4).  Hence,  if  we  find  the  logarithm 
of  the  dividend,  and  from  it  subtract  the  logarithm  of  the 
divisor,  the  remainder  will  be  the  logarithm  of  the  quotient. 

This  additional  caution  may  be  added.  The  difference 
of  the  logarithms,  as  here  used,  means  the  o^g4'>:'aic  differ- 
ence;  so  that,  if  the  logarithm  of  the  divisor  have  a  nega- 
tive characteristic,  its  sign  must  be  changed  to  positive, 
after  diminishing  it  by  the  unit,  if  any,  carried  in  the  sub- 
traction from  the  decimal  part  of  the  logarithm.  Or,  if 
the  characteristic  of  the  logaiithm  of  the  dividend  is  nega- 
tive, it  must  be  treated  as  a  negative  number. 

EXAMPLES. 

1.  To  divide  24163  by  4567. 

log  24163  =  4.883151 
locr    4567  =  3.659631 


Quotient,  5.29078     .     .     0.723520 

2.  To  divide  0.06314  by  .007241. 

log    0.06314  =  2.800305 
log  0.007241  =  3.S59799 


Quotient,  8.7198     .     .     0.94050(5 

Here,  1  carried  from  the  decimal  part  to  the  3,  changes 
it  to  2,  which  being  taken  from  2,  leaves  0  for  the  cha- 
racteristic. 

3.  To  divide  37.149  by  523.76. 

log  37.149  =  1.569947 
log  523.76  =  2.719133 


Quotient,     0.0709274     .     2.S50814 


INTRODUCTION.  2(33 


4.  To  divide  0.7438  by  12.9-176. 

log    0.7-138-1.871456 
locr  12.9476  =  1.112189 


o 


Quotient,  0.057447  .  .  2.759267 

Here,  the  1  taken   from  1,  gives   2   for  a   result,  as  set 
down. 


ARITHMETICAL   COMPLEMENT. 

13.  The  Arithmetical  comj)hment  of  a  logarithm  is  the 
number  which  remains  after  subtracting  the  logaritlim 
from  10. 

Thus,  10  -  9.274687  =  0.725313. 

Hence,  0.725313  is  the  arithmetical  complement 

of  9.274687. 

14.  We  will  now  show  that,  the  difference  heiiceen  tivo 
logaritlims  is  truly  found^  by  adding  to  the  first  loyaritlmi  tlic 
arithnieticcd  complement  of  the  logaritlim  to  he  subtracted^  and 
then  diminishing  the  sum  by  10. 

Let     a  =  the  first  losrarithm, 

b  =  the  logarithm  to  be  subtracted, 
and  c=10— i  =  the  arithmetical  complement  of  ^. 

Now  the  difference  between  the  two  logarithms  will  be 
expressed  by  a  —  b. 

But,  from  the  equation  c  =  10  —  Z^,  we  have 
c-10=-b, 
hence,  if  we  place  for  —  b  its  value,  Ave  shall  have 

a  —  b  =  a  +  c— 10, 
which  agrees  with  the  enunciation. 

When  Ave  Avish  the  arithmetical  complement  of  a  loga- 
rithm, Ave  may  Avrite  it  directly  from  the  table,  by  subtract- 
ing  the  left  hand  figure  from  9,  tlten  proceeding  to  the  right, 
sid)tract  each  figure  from  9  till  loe  reach  tlic  last  figure^  u/tich 
must  be  taken  from  10  :  this  ivill  be  the  same  as  taking  the 
hgaritlim  from  10. 


264  PLANE    TRIGOXOMETRT. 

EXAMPLES. 

1.  From  8.274107  take  2.10-1729. 

By  common  method.  By  arith.  camp, 

3.274107  3.274107 

2.104729        its  cor.  comp.         7.895271 

Diff.       1.169378  Sum  1.169378  after  sub- 

tractins^  10. 

Hence,  to  perform  division  by  means  of  the  arithmetical 
complement,  we  have  the  following 

RULE. 

To  tlie  logarithm  of  the  dividend  add  the  aritlimetical  com- 
plement of  the  logarithm  of  die  divisor:  the  sum,  after  sub- 
tracting  10,  luill  he  the  logarithm  of  the  quotient 

EXA}i[PLES. 

1.  Divide  827.5  by  22.07. 

log  827.5 2.515211 

log  22.07       ar.  comp.       8.656198 

Quotient,  14.839     .     .     .     1.171409 

2.  Divide  0.7488  by  12.9476. 

log    0.7488     ....     1.871456 
log  12.9476     ar.  comp.     8.887811 

Quotient,      0.057447     .     .     .     2.759267 

In  this  example,  the  sum  of  the  characteristics  is  8, 
from  which,  taking  10,  the  remainder  is  2. 

3.  Divide  37.149  by  523.76. 

log  37.149      ....     1.569947 
log  523.76      ar.  comp.      7.280867 

Quotient,         0.0709273     .     .     2.850814 
Divide  0.875  bv  25.  Ans.  0.035. 


INTEODUCTION.  265 

FINDING  THE  POWERS  AND  ROOTS  OF  NUMBERS  BY  LOGARITHMS. 

15.  We  have  (Art.  8), 

10'" -Ji: 
Raising  both   members  of    this  equation  to  the  ?ith  power, 
we  have, 

in  wliich  m  X  n  is  the  logaritlim  of  J/"  (Art.  1)  :    hence, 

The  logarithm  of  any  jpower  of  a  given  number  is  equal  to 
Hie  logaritlim.  of  tlie  number  multiplied  by  the  exponent  of  the 
power. 

16.  Taking  the  same  equation, 

10'"  =  J^ 

and  extracting  the  n\h.  root  of  both  members,  Ave  have 

m  \ 

10"  =J/"' 
2 

in  which  -  is  the  logarithm  of  J/"  :    that  is, 

n 

The  logarithm  of  the  root  of  a  given  number  is  equal  to  thn 
logarithm,  of  the  number  divided  by  the  index  of  the  root. 

EXAMPLES. 

1.  What  is  the  5th  power  of  9  ? 

Log  9  =  0.9542-13;  0.951243x5  =  4.771215; 
whole  number  answering  to  4.771215  is  59049. 

2.  What  is  the  7th  power  of  8  ?  Ans.  2097152 

3.  What  is  the  cube  root  of  4096  ? 

Log  4096  =  3.612360;  3.612360-^3  =  1.204120; 
number  answering  to  1.204120  is  16. 

4.  What  is  the  4th  root  of  .0^000081  ? 

Log  .00000081  =  7.908485  ; 
But,  7.908485  =  8  +  1.908;1S5  ; 

and,  8  +  1.908485  -r-  4  =  2.477121, 

the  number  answering  to  which  is  .03,  which  is  the  roo» 

When  the  characteristic  of  the  logarithm  is  negative^  and  n^i 
divisible  by  the  index  of  the  root,  add  to  it  such  a  negative  number 
as  will  make  the  sum  exactly  divisible  by  the  index,  and  tlifn* 
prefix  the  same  numh2r  to  the  first  decimal  figure  of  the  logarithm. 

5.  What  is  the  6th  root  of  .0432  ?'         Ans.  .592353  -f- 

6.  What  is  the  7th  root  of  .0004967?       Ans.  .3372969, 


266 


PLANE    TRIGONOMETRY. 


GEOMETRICAL   COXSTRUCTIOXS. 


17.  Before  explaining  tlie  method  of  constructing  geo- 
metrical problems,  we  shall  describe  some  of  the  simpler 
ip^Ti-uments  and  their  uses. 


DIVIDERS. 


18.  The  dividers  is  the  most  simple  and  useful  of  the 
instruments  used  for  drawing.  It  consists  of  two  legs  ba^ 
bcj  which  may  be  easily  turned  around  a  joint  at  0. 

One  of  the  principal  uses  of  this  instrument  is  to  lay 
off  on  a  hue,  a  distance  equal  to  a  given  line. 

For  example,  to  lay  off  on   CD  a  distance  equal  to  AB. 

For  this  purpose,  place  the  forefin- 
ger   on   the  joint  of  the  dividers,  and    ^i ,3 

set   one   foot  at  A:   then  extend,  with 

the     thumb     and     other     fingers,    the      k ig j^ 

other  leg  of  the  dividers,  until  its  foot  reaches  the  point 
B.  Then  raise  the  dividers,  place  one  foot  at  C,  and 
mark  with  the  other  the  distance  CB:  this  will  evidently 
be  equal  to  AB. 


RULER   AXD   TRIANGLE. 


19.  A  Ruler  of  convenient  size,  is  about  twenty  inches 
in  length,  two  inches  wide,  and  a  fifth  of  an  inch  in  thick- 


INTEODUCTION.  267 

ness.      It   sliould   be    made   of  a   hard    material,    perfectly 
straight  and  smooth. 

The  hypothenuse  of  the  right-angled  triangle,  which  is 
used  in  connection  with  it,  should  be  about  ten  inches  in 
length,  and  it  is  most  convenient  to  have  one  of  the  sides 
considerably  longer  than  the  other.  We  can  solve,  with 
the  ruler  afid  triangle,  the  two  following  problems. 

I.   To  draw  thron/jh  a  given  point  a  line  which  shall  be  paral- 
lel to  a  given  line. 

20.  Let   C  be  the  given  point,  and  AB  the  given  line. 

Place   the    hypothenuse    of   the    tri-  O 

angle    against    the    edge    of   the    ruler, 

and  then   place   the   ruler   and   triangle     , 

on    the    paper,    so    that    one     of     the  ^ 

sides  of  the  triangle  shall  coincide  exactly  with  AJ3 :    the 
triangle  being  below  the  line. 

Then  placing  the  thumb  and  fingers  of  the  left  hand 
firmly  on  the  ruler,  slide  the  triangle  with  the  other  hand 
along  the  ruler  until  the  side  which  coincided  with  AB 
reaches  the  point  0.  Leaving  the  thumb  of  the  left  hand 
on  the  ruler,  extend  the  fingers  upon  the  triangle  and  hold 
it  firmly,  and  with  the  right  hand,  mark  w4th  a  pen  or 
pencil,  a  line  through  C:   this  line  will  be  parallel  to  AB. 


II.   To  draw  through  a  given  point  a  line  ivhich  shall  he  per- 
jjendicular  to  a  given  line. 

21.  Let  ^^  be  the  given  line,  and  D  the  given  point. 

Place    the    hypothenuse   of  the   tri- 
angle against  the  edge  of  the  ruler,  as 

before.        Then    place    the     ruler     and         I 

triangle    so    that   one   of  the    sides   of       ^ 
the    triangle    shall    coincide    exactly    with    the    line    AB. 
Then    slide    the    triangle    along   the   ruler   uriiil   the   other 
side   reaches   the   point  D:   draw  through   Z^   a   right  line, 
and  it  will  be  perpendicular  to  AB. 


268 


PLANE    TRIGONOMETRY. 


SCALE   OF   EQUAL   PARTS. 


.i^.5    G  .7    S  .970 


22.  A  scale  of  equal  parts  is  formed  by  dividing  a  line 
of  a  given  length  into  equal  portions. 

Tf,  for  example,  the  line  ah  of  a  given  length,  say  one 
inch,  be  divided  into  any  number  of  equal  parts,  as  10, 
the  scale  thus  formed,  is  called  a  scale  of  ten  parts  to  the 
inch.  The  line  a6,  which  is  divided,  is  called  the  unit  of 
the  scale.  This  unit  is  laid  off  several  times  on  the  left 
of  the  divided  line,  and  the  points  marked  1,  2,  8,  &c. 

The  unit  of  scales  of  equal  parts,  is,  in  general,  either 
an  inch,  or  an  exact  part  of  an  inch.  If,  for  example,  a6, 
the  unit  of  the  scale,  ^vere  half  an  inch,  the  scale  would 
be  one  of  10  parts  to  half  an  inch,  or  of  20  parts  to  the 
inch. 

If  it  were  required  to  take  from  the  scale  a  line  equal 
to  two  inches  and  six-tenths,  place  one  foot  of  the  dividers 
at  2  on  the  left,  and  extend  the  other  to  .6,  which  marks 
the  sixth  of  the  small  divisions :  the  dividers  will  then 
embrace  the  required  distance. 

DIAGONAL   SCALE    OF   EQUAL   PARTS. 


dj"                                     c 

\  \  \  \  {  \  \  \  \  i 

»» 

o;) 

\  \  \  \  \  \  \  \  \  \ 

.08 

1  n  U9U  1  I  I 

•07 

n  1  n  I  n  \  I 

•Ofi 

I  I  I  I  I  I  I  I  I  j 

/ 

05 

i    I  hi  III 

1                                                 -^^ 

1     III 

1                                               .03 

11       1  n  1  1 

1                                             nz 

MM       1 

.0, 

nil    u  1  i  I 

i 

£ 

/ 

.1  .2  .3.1  .5.G.7.S  .9    ff 

23.  This  scale  is  thus  constructed.  Take  ah  for  the 
unit  of  the  scale,  which  may  be  one  inch,  ^,  j  or  J  of  an 
inch,  in  length.  On  ah  describe  the  square  ahcd.  Divide 
the  sides  ah  and  dc  each  into  ten  equal  parts.  Draw  af 
and  the  other  nine  parallels  as  in  the  fioure. 

Produce  ha  to  the  left,  and  lay  off  the  unit  of  the 
ricale  any  convenient  nnmber  of  times,  and  mark  the  points 


INTRODUCTION.  269 

1,  2,  o,  &c.  TL(.^n,  divide  the  line  ad  into  ten  equal  parts, 
and  tlirough  the  points  of  division  draw  parallels  to  aZ>,  as 
in  the  figure. 

Now,  the  small  divisions  of  the  line  ah  are  each  one- 
fcenth  (.1)  of  ah;  they  are  therefore  .i  of  ac7,  or  .1  of  ag 
or  gh. 

If  we  consider  the  triangle  acZ/J  we  see  that  the  base  df 
is  one-tenth  of  ad^  the  unit  of  the  scale.  Since  the  distance 
from  a  to  the  first  horizontal  line  above  a5,  is  one-tenth  of 
the  distance  ad,  it  follows  that  the  distance  measured  on  that 
line  between  ad  and  af  is  one-tenth  of  df:  but  since  one-tenth 
of  a  tenth  is  a  hundredth,  it  follows  that  this  distance  is 
one  hundredth  (.01)  of  the  unit  of  the  scale.  A  like  dis- 
tance measured  on  the  second  line  will  be  two  hundredths 
(.02)  of  the  unit  of  the  scale ;  on  the  third,  .03  ;  on  the 
fou:rth,  .01,  &c. 

(f  it  were  required  to  take,  in  the  dividers,  the  anit  of 
th^-  scale,  and  any  number  of  tenths,  place  one  foot  of  the 
dividers  at  1,  and  extend  the  other  to  that  figure  between 
a  and  h  which  designates  the  tenths.  If  two  or  more 
units  are  required,  the  dividers  must  be  placed  on  a  point 
of  division  further  to  the  left. 

When  units,  tenths,  and  hundredths,  are  required,  place 
one  foot  of  the  dividers  where  the  vertical  line  through 
the  point  which  designates  the  units,  intersects  the  line 
which  designates  the  hundredths :  then,  extend  the  dividers " 
to  that  line  between  ad  and  6c  which  designates  the  tenths: 
the  distance  so  determined,  will  be  the  one  required. 

For   example,   to   take   off  the   distance   2.34,  we   place 
one   foot   of  the  dividers  at  I,  and  extend  the  other  to  e 
and  to  take  off  the  distance  2.58,  we  place  one  foot  of  the 
dividers  at  p  and  extend  the  other  to  q. 

Eemark  I.  If  a  line  is  so  long  that  the  whole  of  it 
cannot  be  taken  from  the  scale,  it  must  be  divided,  and 
the  parts  of  it  taken  from  the  scale  in  succession. 

Remark  II.  If  a  line  be  given  upon  the  paper,  ita 
length  can  be  found  by  taking  it  in  the  dividers  and  ap- 
plying it  to  the  scale. 


270  PLANE    TRIGONOMETKY. 


SEMICIRCULAR   PROTRACTOR. 
C. 


A^  B 

24:.  This  instrument  is  used  to  lay  down,  or  protract 
angles.  It  may  also  be  used  to  measure  angles  included 
between  lines  already  drawn  upon  paper. 

It  consists  of  a  brass  semicircle,  ABOj  divided  to  half 
degrees.  The  degrees  are  numbered  from  0  to  180,  both 
ways;  that  is,  from  JL  to  ^  and  from  B  to  A.  The  di- 
visions, in  the  figure,  are  made  only  to  degrees.  There 
is  a  small  notch  at  the  middle  of  the  diameter  AB,  which 
indicates  the  centre  of  the  protractor. 

To  lay  off  an  angle  with  a  Protractor. 
25.  Place  the  diameter  AB  on  the  line,  so  that  the 
centre  shall  fall  on  the  angular  point.  Then  count  the 
degrees  contained  in  the  given  angle  from  A  towards  B^  or 
from  B  towards  J.,  and  mark  the  extremity  of  the  arc  with 
a  pin.  Remove  the  protractor,  and  draw  a  line  through 
the  point  so  marked  and  the  angular  point:  this  line  wiU 
make  with  the  given  line  the  required  angle. 


( 


PLANE    TRIGONOMETR\ 


DEFINITIONS. 

1.  In  every  plane  triangle  tlicre  are  six  parts :  three 
sides  and  three  angles."*^  These  parts  arc  so  related  tc  each 
other,  tliat  when  one  side  and  any  two  other  parts  are 
given,  the  remaining  ones  can  be  obtained,  either  b}^  geo- 
metrical construction  or  by  trigonometrical  computation. 

2.  Plane  Trigonometry  explains  the  methods  of  com- 
puting the  unknown-  parts  of  a  plane  triangle,  when  a  suf- 
ficient number  of  the  six  parts  is  given. 

3.  For  the  purpose  of  trigonometrical  calculation,  the 
circumference  of  the  circle  is  supposed  to  be  divided  into 
360  equal  parts,  called  degrees ;  each  degree  is  supposed 
to  be  divided  into  60  equal  parts,  called  minutes;  and 
each  minute  into  60  equal  parts,  called  seconds. 

Degrees,  minutes,  and  seconds,  are  designated  respec- 
tively, by  the  characters  °  '  ".  For  example,  ten  degrees^ 
eighteen  minutes^  and  fourteen  seconds^  would  be  written 
10°  18'  14". 

4.  If  two  lines  be  drawn  through  the  centre  of  the 
circle,  at  right  angles  to  each  other,  they  will  divide  the 
circumference  into  four  equal  parts,  of  90°  each.  Every 
right  angle  then,  as  EOA^  is  measured  by  an  arc  of  90°; 
every  acute  angle,  as  BOA^  by  an  arc  less  than  90° ;  and 
every  obtuse  angle,  as  FOA^  by  an  arc  gi'cater  than  90°. 

6.  The  complement  of  an  arc  is 
what  remains  after  subtracting  the 
arc  from  90°.  Thus,  the  arc  EB 
is  the  complement  of  AB.  The 
sum  of  an  arc  and  its  complement 
is  equal  to  90°. 

6.  The  supplement  of  an  arc  is 
what  remains  after  subtracting  the 
arc  from  180°.      Thus,   GF  is  the 


272  PLANE    TRIGONOMETRY. 

supplement  of  tlie  arc  AEF.      Tlie  sum  of  an  arc  and  its 
supplement-,  is  equal  to  180°. 

7.  The  sine  of  an  arc  is  the  perpendicular  let  fall  from 
one  extremity  of  the  arc  on  the  diameter  which  passes 
through  the  other  extremity.  Thus,  BD  is  the  sine  of  the 
arc  AB. 

8.  The  cosine  of  an  arc  is  the  part  of  the  diameter  in- 
tercepted between  the  foot  of  the  sine  and  the  centrcv 
Thus,  OD  is  the  cosine  of  the  arc  AB. 

9.  The  tangent  of  an  arc  is  the  line  which  touches  it  at 
one  extremity,  and  is  limited  by  a  line  drawn  through  the 
other  extremity  and  the  centre  of  the  cii^cle.  Thus,  AC  is 
the  tangent  of  the  arc  AB. 

10.  The  secant  of  an  arc  is  the  line  drawn  from  the 
centre  of  the  circle  through  one  extremity  of  the  arc,  and 
limited  by  the  tangent  passing  through  the  other  extremi- 
ty.     Thus,   OC  is  the  secant  of  the  arc  AB. 

11.  The  four  lines,  BD,  OD,  AC,  DC,  depend  for  their 
values  on  the  arc  AB  and  the  radius  OA  ;  they  are  thu8 
designated : 

sin  AB  for  BD 

cos  AB  for  OD 

tan  AB  for  A  C 

sec  AB  for  OC 

12.  If  ABE  be  equal  to  a  quadrant,  or  90°,  then  EB 
will  be  the  complement  of  AB.  Let  the  lines  ET  and  IB 
be    drawn   perpendicular  to   OE.      Then, 

ET,  the  tangent  of  EB.  is  called  the  cotangent  of  AB; 
IB,  the  sine  of  EB,  is  equal  to  the  cosine  of  AB ; 
OT,  the  secant  of  EB,  is  called  the  cosecant  of  AB, 

In  general,  if  A  is  any  arc  or  angle,  we  have, 
cos      ^  =  sin  (90°-^) 
cot      A  ■=  tan  (90°  -  A.) 
cosec  A  =  sec  (90°  —  A) 


PLANE    TRIGONOMETRY.  278 

13.  If  we  take  an  arc,  ABEF, 
greater  than  90°,  its  sine  will  be 
FII;  OH  will  be  its  cosine;  AQ 
its  tangent,  and  OQ  its  secant. 
But  FlI  is  tlie  sine  of  the  arc  GF^ 
which  is  the  supplement  of  AF^ 
and  Oil  is  its  cosine :  hence,  the 
sine  of  an  arc  is  equal  to  the  sine  of 
its  supplement;  and  the  cosine  of  an 
arc  is  eqtial  to  the  cosine  of  its  supplement.'^ 

Furthermore,  AQ  is  the  tangent  of  the  arc  AF^  and 
OQ  is  its  secant:  GL  is  the  tangent,  and  OL  the  secant 
t>f  the  supy)lemental  arc  GF.  But  since  AQ  is  equal  to 
GL,  and  OQ  to  OL,  it  follows  that,  the  tangent  of  an  arc 
is  equal  to  the  tangent  of  its  supplement;  and  the  secant  of  cm 
arc  is  equal  to  the  secant  of  its  supplement.'^ 

TABLE   OF   NATURAL   SINES. 

14.  Let  us  suppose,  that  in  a  circle  of  a  given  radius, 
tnc  lengths  of  the  sine,  cosine,  tangent,  and  cotangent,  have 
been  calculated  for  every  minute  or  second  of  the  quad- 
rant, anJ  arranged  in  a  table;  such  a  table  is  called  a 
table  of  ^ines  and  tangents.  If  the  radius  of  the  circle  is 
1,  the  table  is  called  a  table  of  natural  sines.  A  table  of 
natural  sines,  therefore,  shows  the  values  of  the  sines,  co- 
sines, tangents,  and  cotangents  of  all  the  arcs  of  a  quad- 
rant, which  is  divided  to  minutes  or  seconds. 

If  the  sines,  cosines,  tangents,  and  secants  arc  known 
for  arcs  less  than  90°,  those  for  arcs  which  are  greater  can 
be  found  from  them.  For  if  an  arc  is  less  than  90°,  it^ 
supplement  will  be  greater  than  90*^,  and  the-  numerical 
values  of  these  lines  are  the  same  for  an  arc  and  its  sup- 
plement. Thus,  if  we  know  the  sine  of  20°,  we  also  know 
the  sine  of  its  supplement  1(30°;  for  the  two  are  equal  to 
cacli  other.  The  Table  of  Natural  Sines  is  not  given,  as 
it  is  much  easier  to  make  the  computations  b}^  the  Table 
which  we  are  about  to  explain. 

*  These  relations  are  betAveen  the  numencal  values  of  the  trigonometi-io;J  linos; 
th«  alfjebraic  signs,  which  they  have  in  th»'  different  quadrants,  are  not  oomidored- 

18 


274  PLANE    TRIGONOMETRY. 

TABLE   OF   L0GABITH:5IIC   SINES. 

15.  In  this  table  are  arranged  tlie  logarithms  of  the 
numerical  values  of  the  sines,  cosines,  tangents,  and  co- 
tangents of  all  the  arcs  of  a  quadrant,  calculated  to  a  ra- 
dius of  10,000,000,000.  The  logarithm  of  this  radius  is  10 
Li  the  fii^st  and  last  horizontal  lines  of  each  page,  are  writ 
ten  the  degrees  whose  sines,  cosines,  &c.,  are  expressed  on 
the  page.  The  vertical  columns  on  the  left  and  right,  are 
columns  of  minutes. 

CASE  I. 

To  finely  in    the   table,    the   logarithmic   sine,  cosine,  tangent^  or 
cotangent  of  any  given  arc  or  angle. 

16.  K  the  angle  is  less  than  45°,  look  for  the  degrees 
in  the  first  horizontal  line  of  the  different  pages :  when  the 
degrees  are  found,  descend  along  the  column  of  minutes,  on 
the  left  of  the  page,  till  jou  reach  the  number  showing  the 
minutes  :  then  pass  along  a  horizontal  line  till  you  come  into 
the  column  designated,  sine,  cosine,  tangent,  or  cotangent,  as 
the  case  maj  be :  the  number  so  indicated  is  the  logarithm 
sought.      Thus,  on  page  37,  for  19°  bb',  we  find, 

sine  19°  bb'  .     .     .     .     9.532312 

cos    19°  bb'  ....     9.973215 

tan    19°  bb'  .     .     ,     .     9.559097 

cot    19°  55'  ...     .  10.440903 

17.  If  the  angle  is  gTcater  than  45°,  search  for  the  de- 
grees along  the  bottom  line  of  the  difterent  pages  :  when  the 
number  is  found,  ascend  along  the  column  of  minutes  on  the 
right  hand  side  of  the  page,  till  you  reach  the  number  express- 
ing the  minutes:  then  pass  along  a  horizontal  line  into  the 
column  designated  tang,  cot,  sine,  or  cosine,  as  the  case  may 
be:  tiie  number  so  pointed  out  is  the  logarithm  required. 

18.  The  column  designated  sine,  at  the  top  of  the  page, 
IS  designated  by  cosine  at  the  bottom ;  the  one  designated 
tang,  by  cotang,  and  the  one  designated  cotang,  by  tang. 

The  angle  found  by  taking  the  degrees  at  the  top  of 
the  page,  and  the  minutes  from  the  left  hand  vertical  column^ 
is  the  complement  of  the  angle  found  by  taking  the  degrees 


PLANE    TKIGONOMETRY.  275 

at  the  bottom  of  tlie  page,  and  the  minutes  from  the  right 
hand  column  on  the  same  horizontal  line  with  the  first. 
Therefore,  sine,  at  the  top  of  the  page,  should  correspond 
with  cosine,  at  the  bottom ;  cosine  with  sine,  tang  with 
cotang,  and  cotang  with  tang,  as  in  the  tables  (Art.  12). 

If  the  angle  is  greater  than  90°,  we  have  only  to  sub 
tract   it   from   180°,  and   take   the   sine,   cosine,  tangent,  or 
cotangent  of  the  remainder. 

The  column  of  the  table  next  to  the  column  of  sines, 
and  on  the  right  of  it,  is  designated  by  the  letter  D. 
This  column  is  calculated  in  the  following  manner. 

Opening  the  table  at  any  page,  as  42,  the  sine  of  24° 
is  found  to  be  9.609313 ;  that  of  24°  01',  9.609597 :  their 
difference  is  284 ;  this  being  divided  by  60,  the  number 
of  seconds  in  a  minute,  gives  4.73,  which  is  entered  in  the 
column  D. 

Now,  supposing  the  increase  of  the  logarithmic  sine  to 
be  proportional  to  the  increase  of  the  arc,  and  it  is  nearly 
so  for  60",  it  follows,  that  4.73  is  the  increase  of  the  sine 
for  1".  Similarly,  if  the  arc  were  24°  20',  the  increase  of 
the  sine  for  V'^  would  be  4.65. 

The  same  remarks  are  applicable  in  respect  of  the 
column  D^  after  the  column  cosine,  and  of  the  column  Z), 
between  the  tangents  and  cotangents.  The  column  i),  be- 
tween the  columns  tangents  and  cotangents,  answers  to 
both  of  these  columns. 

Now,  if  it  were  required  to  find  the  logarithmic  sine 
of  an  arc  expressed  in  degrees,  minutes,  and  seconds,  we 
lave  only  to  find  the  degrees  and  minutes  as  before ;  then, 
multiply  the  corresponding  tabular  difference  by  the  sec- 
onds, and  add  the  product  to  the  number  first  found,  for 
the  sine  of  the  given  arc. 

Thus,  if  we  wish  the  sine  of  40°  26'  28". 

The  sine  40°  26'       ...        .      9.811952 
Tabular  difference     2.47  . 
Number  of  seconds     28  . 


Product,  69.16  to  be  added  69.16 

Gives  for  the  sine  of  40°  26'  28".  9.812021. 


275  PLANE    TKIGONOMETEY. 

The  decimal  figures  at  tlie  right  are  generally  omitted 
in  the  last  result;  but  when  they  exceed  live-tenths,  the 
figure  on  the  left  of  the  decimal  point  is  increased  by  1 ; 
the  logarithm  obtained  is  then  exact,  to  within  less  than 
one  unit  of  the  right  hand  place. 

The  tangent  of  an  arc,  in  which  there  are  seconds,  is 
found  in  a  manner  entirely  similar.  In  regard  to  the  co 
gine  and  cotangent,  it  must  be  remembered,  that  they  in- 
crease while  the  arcs  decrease,  and  decrease  as  the  arcs  are 
increased ;  consequently,  the  proportional  numbers  found 
for  the  seconds,  must  be  subtracted,  not  added. 

EXAMPLES. 

1.  To  find  the  cosine  of  3°  40'  40". 

The  cosine  of  3°  40'      .         .         .         9.999110 

Tabular  difference  .13  . 

Number  of  seconds   40 

Product,  5.20  to  be  subtracted  5.20 

Gives  for  the  cosine  of  3°  40'  40"         9.999105. 


2.  Find  the  tangent  of  37"^  28'  31". 

Ans.  9.884592. 

3.  Find  the  cotangent  of  87°  57'  59". 

^715.  8.550356. 

CASE   IL 

To   find    €ie   degrees,  miriutes^   and   seconds    ansicen'ng    to    any 
given  logarWimic  sine,  cosine,  tangent,  or  cotangent. 

19.  Search  in  the  table,  in  the  proper  column,  and 
if  the  number  is  found,  the  degrees  will  be  shown  either 
at  the  top  or  bottom  of  the  page,  and  the  minutes  in  the 
side  column  either  at  the  left  or  rifrht. 

But,  if  the  number  cannot  be  found  in  the  table,  take 
from  the  table  the  degrees  and  minutes  answering  to  the 
nearest  less  logarithm,  the  logarithm  itself^  and  also  the 
corresponding  tabular  difference.  Subtract  the  logarithm 
taken  from  the  table  from  the  given  logarithm,  annex  two 


PLANE    TRIGONOMETEY.  27/ 

ciphers  to  the  remainder,  and  then  divide  the  remainder 
by  the  tabular  difference :  the  quotient  will  be  seconds, 
and  is  to  be  connected  with  the  degrees  and  minutes  be- 
fore found :  to  be  added  for  the  sine  and  tangent,  and 
subtracted  for  the  cosine  and  cotangent. 

EXAMPLES. 

1.  Find  the  arc  answering  to  the  sine     9.880054 
Sine  49°  20',  next  less  in  the  table      9.879963 


Tabular  difference,  1.81)91.00(50". 

Hence,  the  arc  49°  20'  50"  corresponds  to  the  given  sine 
9.880054. 

2.  Find  the  arc  whose  cotangent  is        10.008688 
cot  44°  26',  next  less  in  the  table     10.008591 

Tabular  difference,  4.21)97.00(23". 

Hence,  44°  26' -  23"  =  44°  25'  37"  is  the  arc  answering 
to  the  given  cotangent  10.008688. 

8.  Find  the  arc  answering  to  tangent  9.979110. 

Ans.  43°  37'  21^'. 

4.  Find  the  arc  answering  to  cosine  9.944599. 

Ans.  28°  19'  45". 

20.  AVe   shall   now  demonstrate   the   principal   theorems 
of  Plane  Trigonometry. 


THEOREM  I.  •» 

The  sides  of  a  plane  triangle  are  proportional  to  the  sines  of 
their  opposite  angles. 

2L  Let  ABC  hQ  a  triangle;    then 

CB    :     GA     ::     sin  A     :     sin  B. 

For,  with  ^  as  a  centre,  and  A  I) 
equal  to  the  less  side  BC,  as  a  rn,- 
dins,  describe  the  arc  DI:  and  with 
i?  as   a   centre   and  the  equal   radius       ^^  FI  '-     F 

BCy   describe   the   arc   C7v,  and  di-aw   DB^  and  CF  perpen- 


278  PLANE    TKIGONOMETKY. 

dicular  to  AB:  now  L>F  is  the  sine  of  the  angle  A^  and 
CF  is  the  sine  of  JJ,  to  the  same  radius  AJJ  or  BC.  But 
bj   similar  triangles, 

AD    :     DF    ::     AC    :     CF, 

But  AD  being  equal  to  ^(7,  we  have 

^(7    :     sin  J.     :  :     AG    :     sin  ^,  or 
BC    :     AC    ::     sin  A     :     sin  B. 

By  comparing  the  sides  AB,  AC,  in  a  similar  manner, 
we  should  find, 

AB    :     AC    ::     sin  G    :     sin  B. 

TIIEOEEM  II. 

In  any  triangle,  the  sum  of  the  two  sides  containing  either 
angle,  is  to  their  difference,  as  the  tangent  of  half  the  sum  of 
tlie  two  other  angles,   to  tlie  tangent  of  half  their  difference. 

22.  Let  ACB\i^  a  triangle:    then  Avill 
AB-VAC  :   AB-AC  :  :    tan  i{C-i-B)  :   tan  1{C-B), 

With  J.    as    a    centre,  and    a      E 
radius  AC,  the   less   of   the    two 
given  sides,  let  the  semicircumfe-      i 
rence  IFCF  be  described,  meeting 
AB  in  /,  and  BA  produced,  in  E. 

Then,  ^^will  be  the  sum  of  the  C--.....-- 'FGII 

sides,  and  BI  their  difference.      Draw   CI  and  AF. 

Since  CAE  is  an  exterior  angle  of  the  triangle  ACB^ 
it  is  equal  to  the  sum  of  the  interior  angles  C  and  B  (Bk. 
L,  Prop.  XXV.,  Cor  6).  But  the  angle  CIE  being  at  the 
circumference,  is  half  the  angle  CAE  at  the  centre  (Bk.  III., 
Prop.  XVIIL) ;  that  is,  half  the  sum  of  the  angles  C  and 
B,  or  equal  to  \{C+B). 

The  angle  AFC=ACB,  is  also  equal  to  ABC+BAF; 
therefore.  BAF=  A  CB-  ABC 

But,  ICF=\{BAF)  =  i^{ACB- ABC),  or  i{C- B). 

With  /  and  C  as  centres,  and  the  common  radius  lOy 
let  the  arcs  CD  and  IG  be  described,  and  draw  the  lines 
CE  and  III  perpendicular  to  IC  The  perpendicular  CE 
will    pass    through  E,  the    extremity  of  the    diameter  lE^ 


PLANE    TEIGONOMETHY, 


279 


since  ihe  right  angle  ICE  must  be 
inscribed  in  a  semicircle. 

But  CE  is  the  tangent  of  CIE 
=  l{C-i-B);  ana  Iff  is  the  tan- 
gent of  ICB=^{C~B\  to  the 
common  radius   CI 

But   since   the   lines    CE  and   ///  are   parallel,  the 
angles  Bill  and  BCE  are  similar,  and  give  the  proportion, 

BE    :     BI    ::     CE    :     Iff,  or 

by  placing   for  BE  and  BI,   CE  and  ///,  their  values,  we 
have 
AB  +  AC   :   AB-AC    ::    tan  i{C-{-B)   :    tan  \{C-B). 


B 

tri- 


TIIEOKEM  III. 

In  any  jilane  triarigh,  if  a  line  is  drawn  from  the  vertical 
angle  perpendicular  to  tlic  lose,  dividing  it  into  two  segments: 
Oien,  the  whole  base,  or  sum  of  the  segments,  is  to  the  sum  of 
the  two  other  sides,  as  the  difference  of  those  sides  to  the  differ- 
ence  of  the  segments. 

23.  Let  BAC  be  a  triangle,  and  AD  perpendicular  to  the 
base ;    then 

BG    :     CA-\-AB    ::     CA-AB    :     CD-DB. 

For,  AB'oBD' ■{- AD' 

(Bk.  IV.,  Prop.  XI.) ; 

and  AO^'  =  W^-^Alf  A 

by  subtraction,       AC~  —  AB"  = 

cd'-bd\ 

But   since   the   difference   of  B         D  0 

the  squares  of^foo  lines  is  equivalent  to  the  rectangle  con- 
tained by  their  sum  and  difference  (Bk.  IV.,  Prop.  X.),  we 
have, 

AG^  -  AB'o{A  C-{-  AB) .  {A  C-  AB) 
and  CD'- -  DB'o{ CD  +  DB). (CD-DB) 

therefore,  {CD-]- DB) .{CD-  DB)  =  {A G+AB). {A G-  AB) 
hence,     CD -{- DB   :   AG+AB   ::    AG-AB   :    GD-DB. 


280  PLANE    TRIGO^^OMETRY, 


THEOKEM   IV. 

III  any  right-anrjled  ^^/c^/ie  triangJe^  radius  is  to  tlie  tangent 
of  either  of  the  acute  angles^  as  tJie  side  adjacent  to  die  side 
opposite. 

2tl:.  Let  CAB  be  tlie  proposed  triangle,  and  denote  the 
radius  by  R:   then  ,. 

R    :     tan   C    :  :     AC    :     AB. 


For,  with    any   radius    as    CD    de-      p 


scribe  the  arc  DII^  and  draw  the  tan-  ^ 

gent  DG. 

From   the   simiLar   triangles    CDG   and    CAB,    we  have, 

OD     :     DG     ::      CA     :     AB',    hence, 
R     :     tan   C    :  :      CA     :     AB. 

By  describing    an    arc   with  B   as    a    centre,  we    could 
show  in  the  same  manner  that, 

R    :     tan  B    :  :     AB    :     AC. 


THEOEEM  V. 

Ill  every  right-angled  plane  triangle,  radius  is  to  the  ''osirhf 
of  either  of  the  acute  angles,  as  the  ]iyp)ot]ienuse  to  tlie  sid 
adjacent. 

25.  Let  ABC  be  a  triangle,  right-angled  at  B:  then 

R     :     cos  A     ::     AC    :     AB.  G 

For,  from  the  point  A  as  a  centre, 
with    a    radius   AD=R,    describe   the 
arc  DF,  which  will  measure  the  angle        ^^^         eF 
A,  and  draw  DB  perpendicular   to  AB:   then  will  AB  be 
the  cosine  of  A. 

The  triangles  ADE  and  ACB,  being  similar,  avc  have, 

AD     :     AE    :  :     AC    :     AB :    that  is, 
R    :    cos  A     ::     AC    :     AB. 

REirARK.  The  relations  between  the  sides  and  angles 
of  plane  triangles,  demonstrated  in  these  five  theorems,  aro 


PLANE    TKIGONOMETEY 


281 


aafficient  to  solve  all  the  cases  of  Plane  Trigonometry 
Cir  the  six  parts  which  make  up  a  plane  triangle,  three 
nmst  be  given,  and  at  least  one  of  these  a  side,  before  the 
others  can  be  determined. 

If  the  three  angles  only  are  given,  it  is  plain,  that  an 
indefinite  number  of  similar  triangles  may  be  constructed, 
the  angles  of  which  shall  be  respectively  equal  to  the 
angles  that  are  given,  and  therefore,  the  sides  could  not  be 
determined. 

Assuming,  with  this  restriction,  any  three  parts  of  a 
triangle  as  given,  one  of  the  four  following  cases  will  al- 
ways be  presented. 

I.  "When  two  angles  and  a  side  are  given. 
II.  When  two  sides  and  an  opposite  angle  are  given. 

III.  When  two  sides  and  the  included  angle  are  given. 

IV.  When  the  three  sides  are  given. 

CASE  I. 
When  two  angles  and  a  side  are  given. 

26.  Add  the  given  angles  togetlier,  and  subtract  their 
sam  from  180  degrees.  The  remaining  parts  of  the  tri- 
angle can  then  be  found  bj^  Theorem  I. 


EXAMPLES. 

1.  In  a  plane  triangle,  ABC^ 
there  are  given  the  angle  A  =  SS'^  07', 
the  angle  B=22°  37',  and  the  side 
^Z?=  408  yards.  Eequired  the  oth- 
er parts. 


GEOMETRICALLY. 

27.  Draw  an  indefinite  straio-ht  line,  AB,  and  from  the 
scale  of  equal  parts  lay  off  AB  equal  to  408.  Then, 
at  A.  lay  off  an  angle  equal  to  58°  07',  and  at  B  fin  angle 
equal  to  22°  37',  and  draw  the  lines  AC  and  BC :  then 
will  ABO  be  the  triangle  required. 

The  angle  C  may  be  measured  with  the  protractor  (see 
page  270),  and  when  so  measu.ed,  will  be  found  equal  to 


282  PLxVXE    TKIGONOMETKY. 

99^  16'.  The  sides  AC  and  BC  may  be  measured  by 
referring  tliem  to  the  scale  of  equal  parts  (see  page  268). 
We  shall  find  ^C=  158.9  and  BC=Sdl  yards. 


TRIGOXOMETRICALLY   BY   LOGAKITHMS. 

To  the  angle  .     .     .     ^  =  58°  07' 
Add  the  ande    .     .     ^  =  22°  37' 


Their  sum,  =  80°  4-1' 

taken  from      .     .     .  180°  00' 


leaves  C     .     .     .     .  99°  16',  of  which,  as  it  ex- 

ceeds 90°,  we  use  the  supplement  80°  44.'. 

To  find  the  side  BC. 

sin  C        99°  16'  ar.  comp.  0.005705 

:     sin    A         58°  07' 9.928972 

::       AB  408 2.610660 


:  BC       351.024     (after  rcjectiDg  10)    2.545337. 

Remark.  The  logarithm  of  the  fourth  term  of  a  pro- 
jX)rtion  is  obtained  by  adding  the  logarithm  of  the  second 
term  to  that  of  the  third,  and  subtracting  from  their  sum 
the  logarithm  of  the  first  term.  But  to  subtract  the  first 
term  is  the  same  as  to  add  its  arithmetical  complement 
and  reject  10  from  the  sum  (Int.  Art.  13) :  hence,  the  arith- 
metical complement  of  the  logarithm  of  the  first  term  added 
to  the  logarithms  of  the  second  and  third  terms,  minas  ten, 
will  give  the  logarithm  of  the  fourth  term. 

To  find  the  side  A  G. 

sin    C        99°  16'  ar.  comp.  0.005705 

sin     i?        22°  37' 9.584968 

AB  408 2.610660 


AC       158.C76 2.201333 


2.   In    a    triangle    ABC^    there    are    given    A  =  38°   25', 
B-  57°  42',  and  AB  =  'iOO:    required  the  remaining  paita. 
Ans.  C=83°53',  i?a=  249.974,  ^67=340.04. 


PLANE    TRIGONOMETRY. 


283 


CASE   II. 

When  two  sides  and  an  oj)posite  angle  are  given. 

28.   In    a    plane   triangle,    ABC^  C 

there  are  given  AC=^  216,  CB=  117, 
the  angle  A  =  22^  37',  to  find  the 
other  parts. 


13-.. 


GEOMETRICALLY. 

29.  Draw  an  indefinite  right  line  ABB' :  from  any 
point,  as  J.,  draw  AC,  making  BAC=22°  37',  and  make 
AC  =216.  With  (7  as  a  centre,  and  a  radius  equal  to  117, 
the  other  given  side,  describe  the  arc  B'B;  draw  B'C  and 
BO:  then  will  either  of  the  triangles  ABC  or  AB'Cj  an- 
swer all  the  conditions  of  the  question. 


TRIGOXOMETRICALLY. 

To  find  the  angle  B. 

BC 

117                 ar.  comp. 

7.931814 

AC 

216                                  .     . 

2  334454 

sin  A 

22°  37' 

.    9.584968 

sin  B  45°  13'  55",  or  ABC  134°  46'  05"    9.851236. 


The  ambiguity  in  this,  and  similar  examples,  arises  in 
consequence  of  the  first  proportion  being  true  for  either 
of  the  angles  ABC,  or  AB'C,  which  are  supplements  cf 
each  other,  and  therefore,  have  the  same  sine  (Art.  13). 
As  long  as  the  two  triangles  exist,  the  ambiguity  will  con- 
tinue. But  if  the  side  CB,  opposite  the  given  angle,  is 
greater  than  AC,  the  arc  BB'  will  cut  the  line  ABB',  on 
the  same  side  of  the  point  A,  in  but  one  point,  and  then 
there  will  be  only  one  triangle  answering  the  conditions. 

If  the  side  CB  is  equal  to  the  perpendicular  Cd,  the 
arc  BB'  will  be  tangent  to  ABB',  and  in  this  case  also 
there  will  be  but  one  triangle.  When  CB  is  less  than  the 
perpendicular  Cd,  the  arc  BB'  will  not  intersect  the  base 
ABB  and  in  that  case,  no  triangle  can  be  formed,  or  it 
will  be  impossible  to  fulfil  the  conditions  of  the  problem. 


28-i  PLANE    TRIGONOMETRY. 

2.  Given  two  sides  of  a  triangle  50  and  40  respectively, 
and  the  angle  opposite  tlie  latter  equal  to  32°  :  required 
the  remaining  parts  of  the  triangle. 

Ans.  If  the  angle  opposite  the  side  50  is  acute,  it  is 
equal  to  41°  28'  59" ;  the  third  angle  is  then  equal  to 
106°  81'  01",  and  the  third  side  to  72.363.  If  tlie  ai^gle 
opposite  the  side  50  is  obtuse,  it  is  equal  to  1CS°  31'  (  I", 
the  third  angle  to  9°  28'  59",  and  the  remaining  side  to 
12.436. 

CASE    III. 
When  the  two  sides  and  their  included  an.jle  a^'f.  yzven. 

30.  Let  ABC  be  a  triangle  ;  AB,  a 

BC,    the    given     sides,    and    B    the 
given  angle. 

Since  B  is   known,  Tve   can   find 
the    sum   of  the   two    other  an2;les 
for 

^  +  (7=  180°  -  J?,  and, 
i{A  +  C)  =  i{lSO''-B). 

"We  next  find  half  the  diiference  of  the  angles  A  and 
C  by  Theorem  IL,  viz., 

BC+BA    :    BC-BA     ::    tan  K^l  +  (7)    :    tan  i(^4  -  (7), 

in  which  we  consider  BC  greater  than  BA,  and  therefore 
A  is  greater  than  C\  since  the  greater  angle  must  be  op- 
posite the  greater  side. 

Having  found  half  the  diiference  of  A  and  (7,  by  add- 
ing it  to  the  half  sum,  \[A  +  C),  we  obtain  the  greater 
angle,  and  by  subtracting  it  from  half  the  sum,  we  obtain 
the  less.      That  is, 

\{A+C)^-\{A-C)  =  A,  and 

'M+C)-\{A-c)=a 

Having  found  the  angles  A  and  C.  the  third  side  AC 
may  be  found  by  the  proportion, 

sin  zl     :     sin  ^    :  :     BC    :     A  C. 


PLANE    TRIGONOAIETKY.  285 


EXAMPLES. 


1.  In  tiie  triangle  ABC,  let  BC=54:0,  AB  =  4:60,  and 
the  incluled  angle  J5  =  80°:    required  the  remaining  parts 

GEOMETRICALLY. 

31.  Draw  an  indefinite  right  line  BC,  and  from  any 
point,  as  B,  lay  off  a  distance  BC=54:0,  At  B  make  tlie 
angle  CBA  =  80°  :  draw  BA,  and  make  the  distance 
BA  =  450  ;  draw  A  C ;  then  will  ABC  be  the  required  tri- 
angle. 

TRIGOXOMETRICALLY. 

BG+  BA  =  540  +  450  =  990;  and  BC-BA  =  540  -  450  =  90. 
A+C=  180°  -B=  180°  -  80°  =  100°,    and    therefore, 
i(^  +  (7)  =  i(100°)  =  50°. 

To  find  1{A  -  C). 

BC+BA      990  ar.  comp.  7.004865 

BC-BA        90 1.954243 

::    tan  1{A+C)        50° 10.076187 

:      tan  ^{A  -  C)         6°  11' 9.034795. 

Hence,  50°  +  6°  11' =  56°  11'  =  ^;  and  50° -6°  11'  = 
43°  49'  =  a 

To  find  the  third  side  A  C, 

sin   C        43°  49'  ar  comp.  0.159672 

.     sin    i?        80°        9.993351 

:  :        ^i?      450  2.653213 

:  AC      640.082 2T8062^f6. 

2.  Given  two  sides  of  a  plane  triangle,  1686  and  960, 
aud  their  included  angle  128°  04':  required  the  other  parts. 

Ans.  Angles,  33°  34' 39";    18°  21' 21";    side  2400. 

CASE  IV. 

32.  Having  given  the  three  sides  of  a  pkne  triangle, 
to  find  the  angles. 


286 


PLANE    TEIGONOMETRY. 


Let  fall  a  perpendicular  from  the  angle  opposite  the 
greater  side,  dmding  the  given  triangle  into  two  right- 
angled  triangles :  then  find  the  difference  of  the  segments 
of  the  base  by  Theorem  LEL  Half  this  difference  being 
added  to  half  the  base,  gives  the  greater  segment ;  and, 
being  subtracted  from  half  the  base,  gives  the  less  segment. 
Then,  since  the  greater  segment  belongs  to  the  right-angled 
triangle  having  the  greater  hjpothenuse,  we  have  two 
sides  and  the  right  angle  of  each  of  two  right-angled  tri- 
angles, to  find  the  acute  angles. 

EXAMPLES. 


triangle 


1.  The  sides  of  a  plane 

viz.,  ^(7=40,  J  (7=  84, 
and  AB  =  25  :   required  the  angles. 


being  given 


GEOMETRICALLY. 


83.  With  the  three  given  lines  as  sides  construct  a  tri- 
angle as  in  Prob.  IX.  Then  measure  the  angles  of  the 
triangle  either  ^viih  the  protractor  or  scale  of  chords. 


BC 

That  is. 

Then, 
And, 


TEIGOXOMETEICALLY. 

AO+AB    ::     AC-AB    :     CD-BD, 
40     :     59     ::     9     :     ^^^  =  18.275. 
40  +  18.275 


40 
26.6875  r.  CD. 


40-13.275 


13.3625  =  BD, 


In  the  triangle  BAG,  to  find  the  angle  DAC, 

AC        84  ar.  comp.  8.468521 

DC        26.6375 1.425493 

sin  i)        90° .     .  10.000000 

^mDAC        51°  34' 40"      .     .    ,  .     9.894014. 


PLANE    TRIGONOMETIiY. 


287 


In  the  iriangh  BAD,  to  find  the  angle  BAD, 

AB        25  ar.  comp.  8.602060 

BD        13.3625 1.125887 

sin  Z>        90" 10.000000 

sin  BAD        32°  18'  35" 9.7279-47. 


Hence,  90°  -  I)AC--=  90°  -  51°  34'  40"  =  38°  25'  20"  =  0, 
and,      90°  -  BAD  =  90°  -  32°  18'  35"  =  57°  41'  25"  =  B, 
and,  BAB  +  nAC=-  51°  34'  40"  +  32°  18'  35"  =  83°  53' 
15 "  =  A. 

2.  In    a    triangle,  of  which    the    sides    are  4,  5,  and    6, 
what  are  the  angles? 

Ajis.  41°  24'  35";   55°  46'  16";    and  82°  49'  09". 


SOLUTION   OF   RIGHT-ANGLED   TRIANGLES. 

34.  The  unknown  parts  of  a  right-angled  triangle  may 
be  found  by  either  of  the  four  last  cases ;  or,  if  two  of  the 
sides  are  given,  by  means  of  the  property  that  the  square 
of  the  hypothenuse  is  equivalent  to  the  sum  of  the  squares 
of  the  two  other  sides.  Or  the  parts  may  be  found  by 
Theorems  TV.  and  V. 


EXAMPLES. 

1.  In  a  right-angled  triangle 
BAC,  there  are  given  the  hypothe- 
nuse BC=  250,  and  the  base  AG= 
240 :    required  the  other  parts. 

An^.  B  =  73°  44'  23" ;    C=  16°  15'  37"  ]  AB=  70.0003. 

2.  In    a   right-angled    triangle    BAC,    there    are    given 
Af7  =  384,  and  ^  =  53°  08':    required  the  remaining  parts. 

Ans.  AB  =  287 M;  ^(7  =  479.979;   (7  =  36°  52'. 


288  PLANE    TK  IGON  OAlETll  Y  . 

APPLICATIOX   TO    HEIGHTS   AXl)    DISTANCES. 

1.  A  lIoKizoxTAL  Plane  is  one  Avhicli  is  parallel  to 
the  water  level. 

2.  A  plane  wliicli  is  perpendicular  to  a  horizontal  j^lane, 
is  called  a  vertical  plane. 

3.  All  lines  parallel  to  the  water  level,  are  called  Jiori- 
iontal  lines. 

4.  All  lines  which  are  perpendicular  to  a  horizontal 
plane,  are  called  vertical  lines ;  and  aU  lines  which  are  in- 
clined to  it,  are  called  oblique  lines, 

0.  A  Horizontal  Ajn^gle  is  one  whose  sides  are  hori- 
zontal. 

6.  A  Vertical  Angle  is  one,  the  plane  of  whose  sides 
is  vertical. 

7.  An  angle  of  elevation,  is  a  vertical  angle  having  one 
of  its  sides  horizontal,  and  the  inclined  side  above  the 
horizontal  side. 

8.  An  angle  of  depression,  is  a  vertical  angle  having  one 
of  its  sides  horizontal,  and  the  inclined  side  under  the 
horizontal  side. 

I.  To  determine  the  horizontal  dlMance  to  a  point  ichich  is  tVi- 
accessible  by  reason  of  an  intervening  river. 
35.  Let  C  be  the  point.  Measure 
along  the  bank  of  the  river  a  hori- 
zontal base  line  AB,  and  select  the 
stations  A  and  B,  in  such  a  man- 
ner that  each  can  be  seen  from  the 
other,  and  the  point  C  from  both 
of  them.  Then  measure  the  hori- 
zontal angles  CAB  and  CBA  with 
an  instrument  adapted  to  that  purpose. 

Let  us  suppose  that  we  have  found  AB  =  600  yards, 
CAB  =  57°  35',  and  CBA  =  64"  51'. 

The  angle  (7=  180°  -{A  +  B)=-  57°  34'. 
To  find  the  distance  BC. 
sin  C    57°  34'  '      .  ar.  comp.  .      0.073649 

:     sin  A    57°  35' 0.926431 

:  :        ^15    600 2.778151 

BC   600.11  vards        ....      2.778231 


PLANE    TRIGONOMETRY.  289 

To  find  tlie  distance  A  C. 

sin  0        57°  34'  ar.  comp.  0.073649 

sin  i?         6-i'  51' 9.95674^1 

AB        600  2.778151 


,?  AG        643.94  yards 2.808544. 

11.   To  determine  tJie   altitude  of  an   inaccessihle   object  above  a 
given  liorizontal  plane, 

FIRST   METHOD. 

Z^.  Suppose  D  to  be  tlie  inac-  "     D 

cessible   object,  and  BC  the  bori-  ^-'-''^'iS. 

zontal  plane   from  which  the  alti-      j.  -I'-CC' .^^-^Zfl.jn 

tude  is  to  be  estimated :   then,  if        \/  /'  y^\-ni 

we   suppose  DC  to   be   a  vertical  \  / yC-^^' 

line,  it  will  represent  the  required  \  /;;'--''' 

altitude.  A 

Measure  any  horizontal  base  line,  as  BA  ;  and  at  the 
extremities  B  and  A,  measure  the  horizontal  angles  CBA 
and  CAB.      Measure  also  the  angle  of  elevation  BBC. 

Then  in  the  triangle  CBA  there  will  be  known,  two 
angles  and  the  side  AB;  the  side  BO  can  therefore  be 
determined.  Having  fouud  BC^  we  shall  have,  in  the 
right-angled  triangle  BBC,  the  base  BC  and  the  angle  at 
the  base,  to  find  the  perpendicular  BC,  which  measures 
the  altitude  of  the  point  B  above  the  horizontal  plane  BC 

Let  us  suppose  that  we  have  found 

BA  =  780  yards,  the  horizontal  angle  CBA  =  4A°  24'; 
the  horizontal  angle  CAB=9Q°  28',  and  the  angle  of  eleva- 
tion  Z>i>*(7=10°43'. 

In  the  triangle  BCA,  to  find  the  horizontal  distance  BC 
Tlie  angle  BCA  =  180"  -  (41°  24'  +  96°  28')  =  42°  08'  =  C 

sin  C        42°  08'  ar.  comp.  0.173369 

sin  A        96°  28' 9.997228 

AB        780  2.892095 


BO       1155.29 3.062692. 

19 


290  PLx\.NE    TEIGOXOMETRY. 

In  the  right-angled  triangle  DBC^  to  find  DC. 

R  ar.  comp.  0.000000 

tan  DBC        10M3' 9.277043 

BC        1155.29 3.002092 


DC       218.6-1 2.339735. 


"Remark  I.  It  might,  at  first,  appear,  that  the  solution 
which  we  have  given,  requires  that  the  points  B  and  A 
should  be  in  the  same  horizontal  pLane ;  but  it  is  entirely 
independent  of  such  a  supposition. 

For,  the  horizontal  distance,  which  is  represented  by 
BA,  is  the  same,  whether  the  station  A  is  oru  the  same 
level  with  5,  abov^e  it,  or  below  it.  The  horizontal  angles 
CAB  and  CBA  are  also  the  same,  so  long  as  the  point  C 
is  in  the  vertical  line  DC.  Therefore,  if  the  horizontal 
line  through  A  should  cut  the  vertical  line  DC^  at  any 
point,  as  E^  above  or  below  (?,  AB  would  still  be  the  hori- 
zontal distance  between  B  and  A^  and  AE^  which  is  equal 
to  A  (7,  would  be  the  horizontal  distance  between  A  and  C. 

If  at  J.,  we  measure  the  angle  of  elevation  of  the  point 
/),  we  shall  know  in  the  right-angled  triangle  DAE,  the 
base  AE^  and  the  angle  at  the  base;  from  which  the  per- 
pendicular DE  can  be  determined. 

37.  Let  us  suppose  that  we  had  measured  the  angle  of 
elevation  DAE^  and  found  it  equal  to  20°  15'. 

First :  Li  the  triangle  BA  (7,  to  find  A  C  or  its  equal  AE. 

sin  C       42°  08'  ar.  comp.  0.173369 

sin  ^        41°  24' 9.820406 

AB        780 2.892095 


AC       768.9        2.885870. 


Li  the  right-angled  triangle  DAE,   to  find  DE. 

R  ar.  comp.  0.000000 

tan  A        20°  15       9.566932 

AE       768.9         2.885870 


BE       283.66       2.452802. 


PLANE    TRIGOXOMETRY, 


291 


Bk^j 


Ko\\',  since  DC  is  less  than 
DE,  it  follows  tliat  tlie  station  B 
is  above  the  station  A.      That  is, 

/;/;-  I)C=  283.66  -  218.64  = 
65.02  =  LV, 
whif.li   expresses   the  vertical   dis- 
tance that  the  station  B  is  above 
tlie  station  A. 

Rp:mark  II.  It  should  be  remembered,  that  the  vertical 
distance  which  is  obtained  by  the  calculation,  is  estimated 
from  a  horizontal  line  passing  tlirough  the  eye  at  the  time 
of  observation.  Hence,  the  height  of  the  instrument  is  to 
be  added,  in  order  to  obtain  the  true  result. 


SECOND   METHOD. 

38.  When  the  nature  of  the  ground  will  admit  of  it- 
measure  a  base  line  AB  in  the  direction  of  the  object  D. 
Then  measure  with  the  instrument  the  angles  of  elevation 
at  A  and  B. 

Then,  since  the  ex- 
terior angle  BBC  is 
equal  to  the  sum  of 
the  nnglcs  A  and  ADB^ 


it  [v 


that  the   an- 


gle ADB  is  equal  to  the  difference  of  the  angles  of  eleva- 
tion at  A  and  B.  Hence,  we  can  find  all  the  parts  of  the 
triangle  ABB.  Having  found  BB^  and  knowing  the  angle 
DBC^  we  can  find  the  altitude  DC. 

This  method  supposes  that  the  stations  A  and  B  are  on 
the  same  horizontal  plane ,  and  therefore  it  can  only  be 
used  when  the  line  AB  is  nearly  horizontal 

Let  us  suppose  that  we  have  measured  the  base  line, 
and  the  two  andes  of  elevation,  and 

f  AB    =975  yards, 
found  J  ^       =15^36', 
[BBC  =  27°  29'; 

required  the  altitude  BC. 


292  PLANE    TRIGOXOMETRY. 


Fii-st :   ADB^  BBC  -  .4  =  27='  29'  -  15°  36'  =  11°  53'. 

In  tlie  trianjle  ABB,  to  find  BB, 

sin  B        ir  53'  ar.  comp.  0.686302 

:      sin  A         15°  36' 9.429623 

.  :        AB        975  2.989005 

BB        1273.3 3.104930. 


In  the  triangle  BBC,  to  find  DC. 

R  ar.  comp.  0.000000 

&m  B        27°  29' 9.664163 

BB        1273.3 3.104930 


BC       687.61 2.769093. 


HI.   To  determine  the   "perpendicular  distance  of  an  ohject  below 
a  gicen  Jiorizontal  plane. 

39.  Suppose  C  to  be  directly 
over  the  given  object,  and  .4  the 
point  through  which  the  horizon- 
tal plane  is  supposed  to  pass.  ^'g^^/^^^^'^^s^v-r- 

Measure  a  horizontal  base  line     ^^/l-:i^/4^^^^^^£: 
AB,  and    at  the    stations  ^l    and     ^^C^^^^^^^^^^ 
B    conceive    the    two    horizontal     '''^-^^l/^^^jiii^^^Wi-^^^      -^n>j^^ 
lines  AC,  BC,  to  be  drawn.     The        -^  ^*CP^^^ 

oblique  lines  from  A  and  B  to  the  object  are  the  hy- 
pothenuses  of  two  right-angled  triangles,  of  which  AC^  BC^ 
are  the  bases.  The  perpendiculars  of  these  triangles  are 
the  distances  from  the  horizontal  lines  AC,  BC,  to  the 
object.  If  we  turn  the  triangles  about  their  bases  AC^ 
BC,  until  they  become  horizontal,  the  object,  in  the  first 
Ciise,  will  fall  at  C%  and  in  the  second  at  C". 

Measure  the  horizontal  angles  CAB,  CBA,  and  also  the 
angles  of  depression  C'AC,   C"BC. 


PLANE    TRIGOXOMETBY.  293 

Let  us  suppose  that  we  have 

'  AB     =672  yarcLj 

BAC  =72°  29' 

found  -{  ABC  =  39°  20' 

C'AC  =  27°  4:9' 

^  C"BC=  19°  10'. 

First:   in    the   triangle    ABC^ 
the    horizontal    angle   lci?=  180°  -  (y1 +  i?)  =  180°  -  lir 
49' =  68°  11'. 


To  find  the  horizontal  distance  A  C. 


sin  G        68°  11' 


sin  B 
AB 

AG 


ar.  comp. 


0.032275 


59°  20' 9.801973 


672 


2.827309 


458.79 2.661617. 


To  find  tlie  horizontal  distance  BG. 

sin   G        68°  11'-  ar.  comp.  0.032275 

sin  A         72°  29' 9.979380 

AB        672  2.827369 

BG        690.28 2.839024. 


In  the  triangle   GA  C\  to  find  CC'. 


R 

tan  G'A  G 
AG 

ar.  comp. 

27°  49' 

458.79        

•     • 

0.000000 
9.722315 
2.1)1')  1617 

242.06        

'  triangle  CBG",  to  find 

ar.  comp. 
19°  10'     . 

CG'. 

CG' 

2.383932 

In  thi 

R 

tan  G"BC 

0.000000 
9  541061 

BG 

CG" 

690.28 

239.93 

•    • 

2.839024 
2.380085 

Ucnce    also,    C(7' -  C(7"  =  242.06  -  239.93  =  2.13    yards, 
which  is  the  height  of  the  station  A  above  station  B. 


2'J4: 


PLANE    TRIGOXOMETRY, 


TROBLEMS. 

1.  Wanting  to  know  the  distance  between  two  inacces- 
sible objects,  wLi''L  lie  in  a  direct  level  line  from  the  bot- 
tom of  a  tower  of  120  feet  in  height,  the  angles  of  depres- 
sion are  measured  from  the  top  of  the  tower,  and  are  ibuod 
to  be,  of  the  nearer  57"^,  of  the  more  remote  25*^  oO' :  re- 
quired the  distance  between  the  objects. 

Ans.  173.656  feet. 

2.  In  order  to  find  the  distance 
between  two  trees,  ^1  and  B,  which 
could  not  be  directl}^  measured  be- 
cause of  a  pool  which  occuj)ied  the 
intermediate  space,  the  distances 
of  a  third  point  C  from  each  of 
them  were  measured,  and  also  the 
included  angle  ACB:   it  was  found  that, 

CB    =  672  yards, 
CA     =588  yards, 
ACB  =  b-o'  -io'; 
required  the  distance  AB.  Ans.  592.967  yaras. 

3.  Being  on  a  horizontal  plane,  and  wanting  to  ascer- 
tain the  height  of  a  tower,  standing  on  the  top  of  an  in- 
accessible hill,  there  were  measured,  the  angle  of  elevation 
of  the  top  of  the  hill  40"*,  and  of  the  top  of  the  tower  51° ; 
then  measuring  in  a  direct  line  180  feet  farther  from  the 
hill,  the  angle  of  elevation  of  the  top  of  the  tower  wa? 
83°  45' ;    required  the  height  of  the  tower. 


Ans.  83.998. 


4.  AYanting  to  know  tbe  hori- 
zontal distance  between  two  inac- 
ccvssible  objects  E  and  IT,  the  fol- 
lowing measurements  were  made. 

AB     =536  yards 
BAW=W^  i6' 
viz.i   WAE=br  40' 
ABE  r-.42'  22' 
^EB\V=-1V  07'; 
required  the  distance  i7  IT. 


Ans.  939.527  j^ardfi. 


PLANE    TEIGONOMETKY 


295 


I*=.=kI, 


5.  TVanting  to  know  the 
horizontal  distance  between 
two  inacessible  objects  A 
and  i>,  and  not  finding  any 
station  from  which  both  of 
them  could  be  seen,  two 
points  G  and  Z^,  were  chosen 
at  a  distance  from  each  other,  equal  to  200  yards ;  from 
the  former  of  these  points  A  could  be  seen,  and  from  the 
latter  i>,  and  at  each  of  the  points  G  and  D  a  staff  was 
set  up.  From  G  a  distance  GF  was  measured,  not  in  the 
direction  BG^  equal  to  200  yards,  and  from  D  a  distance 
DE  equal  to  200  yards,  and  the  following  angles  taken. 


VIZ. 


AFG  =  83° 
^^  =  53° 


00', 
SO', 


BDE=b^°  30', 
^Z;C  =  156°2o' 


{aGF  =  5i°  31',     BED  =  88°  30'. 

Ans.  AB=S4:dAQ7  yards. 


6.  From  a  station  P  th-ere 
can  be  seen  three  objects.  A, 
B  and  G,  whose  distances  from 
each  other  are  known :  viz., 
AB--=  SOO,  AG=  600,  and  BG 
=  400  yards.  Now,  there  are 
measured  the  horizontal  an- 
gles. 

APG=SS°  45'  and  BPG 
=  22*°  30':  it  is  required  to 
find  the  three  distances  PA,  PG  and  PB, 


P~~ 


r  PA  =  710.193  yards. 
Ans.  4PG  =  1042.522 
[  Pi?  =  934.291. 


7.  This  problem  is  much  used  in  maritime  survey- 
ing, for  the  purpose  of  locating  buoys  and  sounding  boats. 
The  trigonometrical  solution  is  somewhat  tedious,  but  it 
may  be  solved  geometrically  by  the  following  easy  con- 
struction. 


200 


PLANE    TEIGONOMETRY. 


A.^^ 


Let  .4,  B,  and  C  be  the 
tliree  fixed  points  on  shore, 
and  P  the  position  of  the 
boat  from  which  the  angles 
APC=Zr^b\  CPB  =22^^60', 
and  APB  =  66°  15',  have  been 
measured. 

'Subtract  twice  A  PC  =-67° 
30'  from  180°,  and  lay  off  at 
A  and  C  two  angles,  CAOj 
A  CO,  each  equal  to  half  the 
remainder  =  b6°  15'.  With 
the  point  0,  thus  determined, 
as  a  centre,  and  OA  or  OC  as  a  radius,  describe  the  cir- 
cumference of  a  cii'cle :  then,  an}'  angle  inscribed  in  the 
segment  APC,  will  be  equal  to  oo°  45'. 

Subtract,  in  like  manner,  twice  CPB=4:D°,  from  180°, 
and  lay  off  half  the  remainder  —  67°  30',  at  B  and  C,  de- 
termining the  centre  $  of  a  second  circle,  upon  the  cir- 
cumference of  which  the  point  P  will  be  found.  The 
required  point  P  will  be  at  the  intersection  of  these  two 
circumferences.  If  the  point  P  fall  on  the  circumference 
described  through  the  three  points  A,  B,  and  (7,  the  two 
auxiliary  circles  will  coincide,  and  the  problem  will  be  in- 
determinate. 


J 


ANALYTICAL 
PLANE   TRIGONOMETRY. 


40.  AVe  have  seen  (Art.  2)  that  Phme  Trigonometry 
explains  the  methods  of  computing  the  unknown  })arts  of 
a  plane  triangle,  when  a  sullicient  number  of  the  six  parts 
is  given. 

To  aid  us  in  these  computations,  certain  lines  were  em- 
ployed, called  sines,  cosines,  tangents,  cotangents,  &;c.,  and 
a  certain  connection  and  dependence  were  found  to  exist 
between  each  of  these  lines  and  the  arc  to  A\4iich  it  be- 
longed. 

All  these  lines  exist  and  may  be  computed  for  every 
conceivable  arc,  and  each  will  experience  a  change  of  value 
where  the  arc  passes  from  one  stage  of  magnitude  to  ano- 
ther. Hence,  they  are  called  functions  of  the  arc ;  a  term 
which  implies  such  a  connection  between  two  varying 
quantities,  that  the  value  of  the  one  shall  always  change 
with  that  of  the  other. 

In  computing  the  parts  of  triangles,  the  terms,  sine,  co- 
sine, tangent,  &c.,  are,  for  the  sake  of  brevity,  applied  to 
angles,  bat  have  in  fact,  reference  to  the  arcs  which  measure 
the  angles.  The  terms  when  applied  to  angles,  without 
reference  to  the  measuring  arcs,  designate  mere  ratios,  as  is 
shown  in  Art.  88. 

41.  In  Plane  Trigonometry,  the  numerical  values  of 
these  functions  were  alone  considered  (Art.  13),  and  the 
arcs  from  which  they  were  deduced  were  all  less  than  180 
degrees.  Analytical-  Plane  Trigonomdry^  explains  all  the 
processes  for  computing  the  unknown  parts  of  rectilineal 
triangles,  and  also,  the  nature  and  properties  of  the  circular 
functions,  together  with  the  mctliods  of  deducing  all  the 
formulas   which  express  relations  between   them. 


298 


A^'A].YTICAL 


42.  Let  C  be  the  centre  of  a  circle, 
and  i>J,  EB^  two  diameters  at  right 
anoles  to  eacli  other — dividinor  the  cir- 
cumference  into  four  quadrants.  Then, 
AB  is  called  the  first  quadrant  :  BD 
tbe  second  quadrant ;  DE  the  third 
quadrant;  and  EA  the  fourth  quadrant, 
ino;    their    vertices    at   C,    and    to    which 


D— 


All  angles  hav- 
we  attribute  the 
plus  sign,  are  reckoned  from  the  line  CA^  and  in  the  direc- 
tion from  right  to  left.  The  arcs  which  measure  these  anglea 
are  estimated  from  A  in  the  direction  to  B,  to  X>,  to  E^  and 
to  A  :    and  so  on. 

43.  The  value  of  any  one  of  the  circular  functions  will 
undergo  a  change  with  the  angle  to  which  it  belongs,  and 
also,  with  the  radius  of  the  measuring  arc.  "When  all  the 
functions  which  enter  into  the  same  formula  are  derived 
from  the  same  circle,  the  radius  of  that  circle  may  be 
regarded  as  unity,  and  represented  b}^  1.  The  circular 
functions  Avill  then  be  expressed  in  terms  of  1  :  that  is,  in 
terms  of  the  radius.  Formulas  will  be  given  for  finding 
their  values  when  the  radius  is  changed  from  unity  to  any 
number  denoted  by  R  (Art.  87). 

44.  We  have  occasion  to  refer  to  but  one  circular  func- 
tion not  already  defined.     It  is  called  the  versed  sine. 

The  vtr.<ed  sine  of  an  arc,  is  that  port  of  the  diameter 
intercepted  betw^ecn  the  point  where  the  measuring  arcs  be- 
gin and  the  foot  of  the  sine.     It  is  designated,  versin. 

45.  The  names  which  have  been  given  of  the  circular 
fuHCtions  (Art.  11)  have  no  reference  to  the  quadrants  in 
which  the  measuring  arcs  may  terininate ;  and  heiice,  are 
e<pally  applicable  to  all  angles. 

Eirst  quadmrd. 

If  CA  =  1 
PM  =  sin  a, 
CM  =  cos  a, 
AT  =  tan  a, 
CT  =  sec  a, 
AM=  ver-sin  a. 


PLANE    TRIGONOMETRY. 


299 


Second  quadrant. 

PM--=  sill  a, 
CM  =  cos  a, 
AT  —  tan  a 
CT  =  sec  a, 
AM  =  ver-sin  a. 

lliird  tjuadrant. 

PM  =  sin  a, 
CM  =  cos  a, 
AT  =  tan  a, 
CT  =  sec  a, 
-4J/=  ver-sin  a. 

Fourth  quadrant. 

PM  =  sin  a, 
6^J/  =  cos  «, 
J.  7^  =  tan  a, 
CT  =  sec  «, 
^J/=  ver-sin  a. 


46.    We  will  noAv  proceed  to  established  some  of  tlie  ini 
portant    general    relations    between   the  rp 

circuUir  functions.  -P/-- ■^^\]^/^ 

Regarding  the  radius   CP  of  the  cir- 
cle as  unity,  and  denoting  it  by  1  (Art.     /    Kl 
43) ;    Ave  have  in  the  right-angled  trian- 
gle CPM, 

Plf  -\-  CTf  =  if  =  1,  ^ 

that  is,        sin    a  +  cos'  a  =  1,  *     .     (1) 

47.   Since  the   triangles  CPM  and    CTA    are  similar,    we 
have, 

A  T       PM 
CA  ~   CM' 


that  is, 


tan  a 


sm  a 


cos  a 


(2) 


*  The  symbols  »\u  «,  cos'^  a,   taa^  a,    &c.,    signify  the   square  of  the  sine,  Ums 
y^uare  of  Ute  cosine,  &c. 


80C 


ANALYTICAL 


48.   Substituting  in  equation  (2),  90  —  a  for  a,  wc  have, 


tan  (90  -  a)  = 


sin  (90  -_g) 

cos  (90  —  a) 


that  is  (Art.  12),    cot  a  = 


cos  a 

sin  a 


(3) 


49.  Multiplying  equations  (2)  and  (3),  member  bj  mem- 
ber, we  have, 

tan  a  X  cot  a  =  1 (4) 

50.  From  the  two  similar  triangles  (7PJ/ and  67"yl,  we 
have, 

CT  _  CP^, 
CA  ~  CM' 

_1_ 

cos  a 

51.  Substituting  for  a,  90  —  a,  we  have, 

1 


that  is. 


(5) 


sec  (90  -  a)  = 


cos  (90 -a) 


that  is,  cosec  a  =  -. -•  .     .     . 

52.   In  the  right-angle   CTA^  we  have, 
C2'-  =  CA.'  +^17'"  ; 
that  is,  sec"  a  =  1  +  tan'  a.     .     . 


(6) 


(7) 


53.  Substituting  (90  — c/)  for  a,  in  equation  (7)  and  recol- 
lecting that  sec  (90— a)=cosec  a,  and  tang  (90— a)  =  cot  a, 
we  have  ^ 


cosec^  a  —  1  +  cot  ^  a. 


54.  We  have,  AM  equal  to  the    versed  sine  of   the  aro 
APy  hence. 


ver-sin  a  =  1  — cos  a. 


(9) 


PLANE    TRIGONOMETRY. 


801 


55.  These  nine  formulas  being  often  referred  to,  we 
sLall  place  them  in  a  table. 

They  are  nsed  so  frequently,  that  they  should  be  coiu- 
niitted  to  memory. 


TABLE   I. 


1. 

2          2 

.  sin"  a  +  cos"  a 

r::r 

le  =  1. 

2. 

• 

• 

tan  a 

= 

sin  a 

cos  a 

3. 

• 

,     cot  a 

= 

cos  a 

sin  a 

4. 

, 

• 

tan  a  X  cot  a 

= 

R'  =  1. 

5. 

. 

.  sec  a 

= 

1 
cos  a 

6. 

. 

'• 

cosec  a 

= 

1 

sin  a 

7. 

. 

2 

.  sec  a 

= 

1  +  tan"  a. 

8. 

, 

, 

2 

cosec"  a 

= 

1  +  cot"  a. 

9. 

• 

.  ver-sin  a 

= 

1  —  cos  a. 

56.  We  will  now  explain  the  principles  whicli  deter- 
mine the  algebraic  signs  of  the  trigonometrical  functions. 
There  are  but  two. 

1st.  All  lines  estimated  from  DA^  vpivards^  are  consid- 
ered positive^  or  have  the  sign  +  :  and  all  lines  estimated 
from  DA^  in  the  opposite  direction,  that  is,  downwards,  are 
considered  negative,  or  have  the  sign  — . 

2(L  All  lines  estimated  from  FB  along  CA,  that  is,  (o 
the  right,  are  considered  positive,  or  have  the  sign  +  :  and 
all  lines  estimated  from  FB  along  CD,  that  is,  in  the  ojipo- 
site  direction,  are  considered  negative,  or  have  the  sign  —. 


302 


ANALYTICAL 


57.  Let  lis  determine,  from  the  above  jirinciples,  the 
a]gebraic  signs  of  the  sines  and  cosines  in  the  dilferent 
quadrants. 

First  quadrant. 

68.   In  the  first  quadrant. 

PM  =  sin  a, 
and  Pm  =  CJI=  cos  a. 


B 


are  loth  positive^  the  former  being  above 
the  line  DA,  and  the  latter  being^  esti- 
mated  from   C  to  the  right  (Art.  56). 

Second  quadrant. 

59.  In  the  second  quadrant, 

PJI  =  sin  a, 
and  Pm  =  CJf  =  —  cos  a  : 

the  sine  is  positive,  being  abov^e  the  line 
DAj  and  the  cosine  negative  being  esti- 
mated to  the  left  of  PK 

Third  quadrant. 

60.  In  the  third  quadrant, 

PM  =  —  sin  rt, 
and  Pm  =  CM  =  —  cos  a  : 

the  sine  is  negative,  fiilling  below  the 
line  PA^  and  the  cosine  is  negative, 
being  estimated  to  the  left  of  the  cen- 
tre a 

Fourth  quadrant. 

61.  In  the  fourth  quadrant, 

PM  =  —  sin  (7, 
and  Pm  =  CM  =  cos  a  : 

the  sine  is  negative,  falling  below  the 
line  PA^  and  the  cosine  is  positive,  fall- 
ing on  the  right  of  FB.  Ilence,  we 
conclude,  that 


D- 


M 


PLANE    TRIGONOMETRY.  303 

Ist^  The  sine  is  positive  in  the  first  and  second  quadrants^  and 
ncrjatice  in  the  tliird  and  foartli  : 

2d.  TJic  cosine  is  positive  in  the  first  and  fourth  quadrants^ 
and  nerjative  in  the  second  and  Lliird : 

In  other  words, 

Ist.  The  sine  of  an  angle  less  titan  180*^  is  positive  ;  and  the 
sine  of  an  angle  greater  than  180^  and  less  than  oGO"^,  is 
rifgaiive  : 

2d.  Tlte  cosine  of  an  angle  less  than  90°  is  positive  ;  the  cosine 
of  an  angle  greater  than  90°,  and  less  than  270°,  is  nega- 
tive; and  the  cosine  of  ayi  angle  greater  titan  270°,  and  less 
than  860^,  is  j^ositive. 

62.  The  algebraic  signs  of  the  sine  and  cosine  being 
determined,  the  signs  of  all  the  other  trigonometrical  func- 
tions may  be  at  once  established  by  means  of  the  formulas 
of  Table  I. 

Thus,  for  example, 

sin  a 


tan  a  = 


cos  a 


Now,  if  the  algebraic  signs  of  sin  a  and  cos  a  are  alike, 
the  tangent  is  positive  ;  if  they  are  unlilvc,  it  is  negative. 
Ilence,  the  tangent  is  positive  in  the  first  and  third  quadrants^ 
and  negative  in  the  second  and  fourth. 

The  same  is  also  true  of  the  cotan^rent :   for, 


63.    Again,  since 


' 

a 

cot 

a 

= 

cos 

a 

sin 

a 

1 

sec 

a 

— 

cos 

— » 
a 

the  sign  of  the  secant  is  always  the  same  as  that  of  the  cosine, 

1 


And  since. 


cosec  a 


sm  a 

the  sign  of  the   cosecant   is    always    Hie   same    as    that   of  the 
sifie. 


801 


ANALYTICAL 


6-i.  The  versed  sine  is  constantly  positive.  For,  il  is 
alwa^'S  found  by  subti-acting  the  cosine  from  radius,  and 
the  remainder  is  a  positive  quantit^^,  since  the  cosine  can 
never  exceed  radius.  AVh,en  the  cosine  is  negative,  the 
versed  sine  becomes  greater  than  radius. 

66.  Let  q  denote  a  quadrant:  then  the  following  table 
will  show  the  algebraic  signs  of  the  trigonometrical  hues 
in  the  different  quadrants. 


sine 
cosine 
tangent 
cotano-ent 


First  q. 
+ 

+ 

+ 


Second  q.     Third  q.     Foirrth  q. 


4- 


+ 


+ 


Q>6.  AVe  have  thus  for  supposed  all  angles  to  be  esti- 
mated from  the  line  CA  from  right  to  left,  that  is  in  the 
direction  from  A  to  B^  to  i),  &;c.,  and 
have  also  regarded  such  angles  as  posi- 
tive. It  is  sometimes  convenient  to 
give  different  signs  to  the  angles  them- 
selves. 

If  we  suppose  the  angles  to  be  esti- 
mated from  CA,  in  the  direction  from 
left  to  right,  that  is,  in  the  direction  from  A  to  E,  to  i>, 
(Sec,  we  must  treat  the  angles  themselves  as  negative,  and 
affect  them  with  the  sign  — . 

For  a  negative  angle  less  than  90°,  the  sine  Avill  be 
negative,  and  the  cosine  positive :  for  one  greater  than  90° 
and  less  than  180°,  the  sine  and  cosine  will  bc'th  be  nega- 
tive. The  algebraic  sign  of  the  sine  always  changes,  when 
we  change  the  sign  of  the  arc,  but  the  sign  cf  the  cosine 
remains  the  same.  Ilence,  calling  x  the  ar3,  we  have  m 
general, 

sin  (  —  a:  )=  —  sin  cc, 
cos  {  —  x)  =       cos  a:, 

tan  {—  x)  =  —  tan  cc, 

cot  {—  x)  =  —  cot  X. 

67.   We    shall    now   examine   the    chansres    which    take 


PLANE    TKIGONOMETRY.  305 

place  in  the  values  of  the  trigonometrical  lines,  as  the 
angle  increases  from  0  to  800°,  ajid  shall  begin  with  the 
jsine  and  cosine. 

When  the  arc  is  zero,  tha  sine  is  0,  and  the  cosine 
equal  to  R  =  1.  At  90°  the  sinie  becomes  equal  to  li  =  1, 
and  the  cosine  becomes  0.  At  180°,  the  sine  becomes  Oy 
and  tlie  cosine  equal  to  —  B  =  —  1.  At  270°,  the  sine 
becomes  equal  to  —  i^  =  —  1,  and  the  cosine  equal  to  0. 
At  8(J0°,  the  sine  becomes  equal  to  0,  and  the  cosine  to 
R  =  1.     llence, 

First  quadrant. 

As   the  arc  increases  from  0  to  90°  : 
The  sine  increases  from  0  to  1  : 
The  cosine  decreases  from  1  to  0. 

Second  quadrant. 

As  the  arc  increases  from  90°  to  180°  : 

The  sine  decreases  from  1  to  0 : 

The  cosine  increases,  numerically,  from  0  to  —  1. 

Third  quadrant. 

As  the  arc  increases  from  180°  to  270°: 

The  sine  increases,  numerically,  from  0  to  —  1  : 

The  cosine  decreases,  numerically,  from  —   1  to  0. 

Fourth  quadrant. 

As  the  arc  increases  from  270°  to  360°: 

The  sine  decreases,  numerically,  from  —  1  to  0 : 

The  cosine  increases  from  0  to  i?  =  1. 

68.  By  a  careful  consideration  of  the  preceding  princi- 
ples and  by  making  the  proper  substitutions  in  the  formulas 
already  deduced,  we  may  now  form  the  following  Table: 


20 


806 


ANALYTICAL 

TABLE    II. 


sin       0               =0, 

sin  (180°  +  a) 

=  —  sin  a, 

cos      0               =1, 

cos (180°  +  a) 

=  —  cos  a, 

tan      0               =0, 

tan (180°  +  a) 

=      .  tan  a, 

cot      0               =00. 

cot (180°  +  a) 

=        cot  a. 

sin    (90°  —  a)  =  cos  a, 

sin  (270°  -  a) 

=  —  cos  a, 

cos    (90°  -  a)  =  sin  a, 

cos  (270°  -  a) 

=  —  sin  a, 

tan    (90°  ~  a)  =  cot  a, 

tan  (270°  -  a) 

=       cot^a, 

cot    (90°  -  a)  =  tan  a. 

cot  (270°  -  a) 

=       tan  a. 

sin     90°             =  1, 

sin    270° 

=  -1, 

cos    90°             =  0, 

cos  270° 

=       0, 

tan    90°             =00, 

tan   270° 

=    —  00, 

cot    90°             =  0. 

cot  270° 

=       0. 

sin    (90°  +  a)  =        cos  a, 

sin  (270°  +  a) 

=  —  cos  a, 

cos    (90°  +  a)  =  —  sin  a, 

cos  (270°  4-  a) 

=        sin  a, 

tan    (90°  +  a)  =  -  cot  a, 

tan  (270°  +  a) 

=  —  cot  a, 

cot    (90°  4-  a)  =  —  tan  a. 

cot  (270°  +  a) 

=  —  tan  a. 

sin  (180°  —  a)  =       sin  a, 

sin  (360°  -  a) 

=  —  sin  a, 

cos  (180°  —  a)  =  —  cos  a, 

cos  (360°  -  a) 

=       cos  a, 

tan  (180°  -  a)  =  -  tan  a, 

tan  (360°  -  a) 

=  —  tan  a, 

cot  (180°  -  a)  =  -  cot  a, 

cot  (360°  -  a) 

=  —  cot  a. 

sin    180°           =        0, 

sin   360° 

=  0, 

cos   180°           =  -  1, 

cos  360° 

=  1, 

tan   180°           =       0, 

tan  360° 

=  0, 

cot   180°           =  -  oo. 

cot  360° 

=  00. 

69.  The  examinations  thus  for,  have  been  limited  to 
arcs  which  do  not  exceed  360°.  It  is  easily  shown, 
however,  that  the  addition  of  360°  to  any  arc  as  x,  will 
make  no  difference  in  its  trigonometrical  functions;  for, 
such  addition  would  terminate  the  arc  at  precisely  the 
same  point  of  the  circumference.  Ilence,  if  0  represent  au 
entire  circumference,  or  360°,  and  ?i  any  whole  number, 
we  shall  have, 

sin  (C  +  a:)  =  sin  a:  ;  or    s^'n  (?i  X  C  -\-  x)  =  sin  x. 
The  same  is  also  true  of  the  other  functions. 


PLANE    TRIGONOMETKY. 


307 


70.  It  will  farther  appear,  that  whatever  be  the  value 
of  an  arc  denoted  by  x,  the  sine  mav  be  expressed  by 
that  of  an  arc  less  than  180°.  For,  in  the  first  place,  we 
may  subtract  360^  from  the  arc  x,  as  often  as  360°  is 
contained  in  it:  then  denoting  the  remainder  by  3/,  we 
hive, 

sin  X  =  sin  y. 

Then,  if  y  is  greater  than  180°,  make 

y  -  180=  =  z, 
and  we  sliall  have, 

sin  y  =  —  sin  2. 
Thus,  all  the   cases  are  reduced   to   that   in  which  the  arc 
whose    functions   we  take,  is  less  than  180°  ;    and  since  we 
also  know  that, 

sin  (90  +  a:)  =  sin  (90  -  x\ 

they  are  ultimately  reducible  to  the   case    of   arcs  between 
0  and  90°. 

GENERAL    FORMULAS. 


71.    To  find   the  formida  for  the  sine  of  the  difference  of  two 
angles  or  arcs. 

Let  ACB  ha  ix.  triangle.  From  the 
vertex  C  let  fall  the  perpendicular  CD^ 
on  the  base  AB^  produced. 

Denote  the  exterior  angle  CBD  by 
a,  and  the  angle  CAB  by  h. 

Then,  AB  =  AD-  DB. 

But  (Art.  25),  AD  =  AC  cos  h,  and  BD  =  BO  cos  CBD, 

Ilence.  AB  =  AC  cos  b  —  BC  cos  a. 

Dividing  both  members  by  AB^  we  have 

,      AC         ^       BC 
l=;^cosZ>--^co3  a. 

But,  since  sin  a  =  sin  CBA,  we  have  (Art.  21) 

AC       sin  a  -.       BC        sin  h 

AB  ^  im~^'       ^"^         AB  """^^(7  ' 


i^OS 


ANALYTICAL 


licnce 


or, 


sill  a  sin  h 

1  =  -T— 7^  cos  h -. — 7^ 

sin  C  sin  C 


cos  a. 


sin  6*  =  sin  a  cos  h  —  sin  Z^  cos  a^ 


But  the  ar.o;le  C  is  equal  to  tlie  difl'erence  between  tlio 
angles  a  and  b  (Geom.  B.  L,  P.  25,  C.  6)  :    hence, 

sin  (a  —  h)  =  sin  a  cos  h  —  cos  a  sin  Z^ ;    .     .  («) 

that  is,  TJie  sine  of  the  d/JP^reiice  of  any  Uco  arcs  or  aDrjles 
is  equal  to  the  sine  of  the  Jir.^t  ijito  the  cosine  of  the  second^ 
iniims  the  cosine  of  iJie  first  into  the  sine  of  the  second. 

It  is  phiin  that  the  fonnula  is 
equally  true  in  whichever  quadrant 
the  vertex  of  the  angle  C  be  placed: 
hence,  the  forinuhi  is  true  for  all 
values,  of  the  arcs  a  and  b. 


r^ 

'~\ 

/ 

>) 

l^ 

c 

^ 

72.  To  find  the  formula  fir  the  sine  of  the  sum  of  two  anjles 

or  arcs. 

Bj  formula  (a) 

sin  {a  —  b)  =  sin  a  cos  b  —  cos  a  sin  b, 

substituting  forZ>,— Z),  and  recollecting  (Art.  66)  that, 
sin  {—  x)  =  —.sin  x 
and  cos  (—  .r)  =  cosx; 

and  also  that     a  —  {—  b)  =  a  +  b, 
we  shall  have,  after  makinnj  the  substitutions  and  combininc^ 
the  algebraic  signs, 

sin  (a  -{-  b)  =  sin  a  cos  b  +  cos  a  sin  b.     .    {b) 

73.  To  find  the  formula  for    the    cosine    of   the  sum   of  tufO 

angles  or  arcs. 

By  formula  (b)  we  have, 

sin  {a  +  b)  =  sin  a  cos  b  +  cos  a  sin  b, 
substitute  for  a,  90°  -f-  a,  and  we  have, 
sin  [(90°  -\-  a)+b]=  sin  (90°  +  a)  cos  b  -f  cos  (90°  +  a)  sin  b. 


PLANE     TiilGOiSOMETKY.  iiOy 

But,      sin  [90°  +  («  +  b)]  =  cos  (a  +  b)    (Table  IL)  ; 

sin  (90''  +  a)  =  cos  a, 
and,  cos  (90^  +  a)  —  —  sin  a  ; 

making  the  substitutions,  we  liave, 

cos  {a  +  b)  =  cos  a  cos  b  —  sin  a  sin  i.     .     .     (c) 

74.  2b  Jind  the  formula  fur  the  cosine  of  Ike  dijjereitce  between 

ttco    uu'jles   or   arcs. 

By  formula  {b)  we  have, 

sin  {a  +  b)  =  sin  a  cos  b  -f  cos  a  sm  h. 
For  a  substitute  90^  —  «,  and  we  have, 
sm  [90°  -  (a  -  b)]  =  sin  (90°  -  a)  cos  b  +  cos  (90°  -  a)  sin  i. 
But,       sin  [90°  -  {a  -  b)]  =  cos  (a  -  b)  (Table  II.), 
sin  (90°  —  a)  =  cos  a, 
cos  (90^  —  a)  =  sin  a  ; 
making  tlie  substitutions,  we  have, 

cos  (a  —  b)  =  cos  rt  cos  b  +  sin  a  sin  b.    .     .     (fZ) 

75.  :7t>  yi?icZ  ^/ie  formula  for   the    tangerd   of  the   sum   of   two 


Bj  Table  L, 

,        /  ,    7x        ^in   {a  +  b) 

tan  (a  +  h)  = ^ r  ' 

^  ^        cos  [a  +    /) 


(c/  +  b) 

sin  a  cos  />  +  cos  a  sin  />     ,       ,,.        ,  ,  , 

= ,    — . -. — r>    by  (b)  and  (c) , 

cos  a  cos  6  —  sin  a  sm  o        -^    ^  ^ 

dividing   both  numerator  and  denominator   by  cos  a  cos  i, 


1    + 7' 

=  cos  a  cos  6       cos  «  cos  6 


sin  a  cos  i  cos  a  sin  Z> 
ji;s  «  cos  6 
sin  a  sin  6 
cos  a  cos  6 


tan  «  -f  tan  /; 


tan  (a  -^  b)  =  ^ 

^  1  —  tan  a  tan 


b 


(/) 


SIO  ANALYTICAL 


76     To  find  the  tangent  of  the  difference  of  two  arcs. 

,       I         ;,       sin  (a  -  h)  (Table  I). 

tau  (a  -  l>)= 7 rf  »  ^  ^ 

^        cos  (a  —  /') 

sio  a  cos  h  —  cos  a  sin  Z>     ,      ,  ,       ,  ,  ., 

=  _       _. ^. —    ,  by  (a)  an  J  ((/). 

cos  a  cos  b  +  sin  a  sin  u       ''      ■' 

Dividing  both  numerator  and  denominator  by  cos  a  cos  6», 

tan  a  —  tan  h  ,. 

tan   (a  -  Z.)  =  .— ^^ r-     •     •     (y) 

^  1  +  tan  a   tan  b 

77.    The  student  will  find  no   diPiiculty  in  deducing  the 

following  formulas. 

.    ,,        cot  a  cot  h  —  1  /7X 

cot  (a  +  Z.    =  ^ -J-  ,     .     .      (A) 

^  cot  a  4-  cot  6/ 

,,        cot  a  cot  /y  +  1  /-v 

cot   (a  -  Z/)  =  ^ .     .      (0 

cot  b  —  cot  a 

78.  7I>  fi.nd   the   sine  of  twice  an  arc^  in  functions  of  the   arc 

and  radius. 
By  formula  (U) 

sin  (a  +  ^)  =  sin  a  cos  b  +  cos  a  sin  Z*. 
Make  a  =  b,  and  the  formula  becomes, 

sin  2a  =  2  sin  a  cos  a.       ...     (k) 

If  we  substitute  for  «,   -—  ?  we  have, 

sin  a  =  2  sin  ^a  cos  Ja.        .     .     (^1) 

79.  7o  find  the  cosine  of  ticice  an  arc  in  functions   of  tJie  arc 

and  radius. 
By  formula  (c) 

cos  {a  +  //)  =  cos  a  cos  b  —  sin  a  sin  b.    ' 
Make  a  =  b.  and  we  have, 

cos  2cf  =  cos-  «  —  sin-  a {!) 

By  Table  I.,   sin-  r/  =  1  —    cos'  a  ;    hence,  by  substitution, 

cos  2a  =  2  cos^  a  -  1.     .     .     .     (11) 
Again,  since  cos-  a  =  1  —  sin-  «,  we  also  have, 

cos  2a  -  1  -  2  sin^  a.     .     .     .     (?  2) 


PLANE    TRIGONOMETRY.  311 

80.    To  determine  the  tangent  of  twice   or  thrice  a  given  arc  in 
functions  of  the  arc  and  radius. 

By  formula  {f) 

tan  a  +  tan  & 
tan  (a  +  h)  =  i^-tan^l^l,* 

Make  h  =  a,  and  we  liave, 

2  tan  a  /«,\ 

tan  2a  =  , ,— o— *    •     •     •     V^; 

1  —  tan-  a 

Making  h  =  2a,  we  have, 

tan  a  +  tan  2a 
tan  8a  =  ^  .,       o    y 

1  —  tan  a  tan  2a 

substituting  the  value  of  tan  2a,  and  reducing,  we  have, 

tan  3a  =  ^-i^",-:f -T-  ;      •    •     (»  D 
1—3  tan-  a 

The  student  will  readily  find 

^  ^         cot  a  —  tan  a  r^^ 

cot  2a  =  — ^ W 


81.    To  find  the  sine  of  half  an  arc  in  terms  of  the  functuyra 
of  the  arc  and  radius. 

By  formula  {I  2) 

cos  2a  =  1  —  2  sin^  a. 
For  a,  substitute  ^a,  and  we  have, 

cos  a  =  1  —  2  sin-  \a ; 
hence,  2  sin-  ^a  =  1  —  cos  a, 


sin  \ 


a-^/lZ^^^.       .     .     (o) 


82.    To  find  the  cosine    of  half  a   given  arc   in   terms    of  the 
functions  of  the  arc  and  radius. 

By  formula  {1 1) 

cos  2a  =  2  cos-  a  —  1. 
For  a,  substitute  J  a,  and  we  have, 

cos  a  =  2  cos-  |a  —  1  ; 


^  /l  +  cos  a  (^\ 

hence,  cos  Ja  =  y n •     •     •     v/'; 


^12  AXALYTICAL 

83.     To  find  the   tangent  of  JiaJf  a  given  arc,  in   /'unctions    of 
the  arc  arid  radius. 

Divide  formula  (o)  b}^  (j?),  and  we  have, 


tan  \ 


a=s/\^^--,      .     .     .     (,) 
^    1  4-  cos  a 


Multiplying  botli  terms  of  the  second  member  by  Vl  —  cos  ti. 

and  reducin.s;  tan  ^a  =  -. ^ —  >    •     •     •     •     (?  1) 

°  "^  sin  a 

Multiplying  both   terms   by  the    denominator    y/ 1  +  cos  a, 


and  reducing  tan  Ja  =  .,    ,  ,    •     •     •     •     (^2) 


sm  a 
1  +  cos  a 


GEXERAL    FORMULAS. 

84.  The  formulas  of  Articles  71,  72,  73,  7-i,  furnish  a 
great  number  of  consequences ;  among  which  it  will  be 
enough  to  mention  those  of  most  frequent  use.  By  adding 
and  subtracting^  ^ve  obtain  the  four  which  follow, 

sin  {a  +  h)  -h  sin  (a  —  h)  =  2  sin  a  cos  h,  .  (r) 

sin  (a  -h  l>)  ^  sin  [a  —  h)  =  2  sin  h  cos  a,  .  (s) 

cos  (a  -\-  I)  +  cos  {a  —  h)  =  2  cos  a  cos  b,  .  {i) 

cos  (a  —  h)  —  cos  {a  -f  Z/)  =  2  sin  a  sin  b^  .  (u) 

and  which  serve  to  change  a  product  of  several  sine?  c>i 
cosines  into  linear  sines  or  cosines,  that  is,  into  sines  an'1 
cosines  multiplied  only  by  constant  quantities. 

85.  If  in  these  formulas  we  put  a  +  b  =  p,  a  —  b  ^  q^ 
which  gives  a  =  ^^—^j  b  =  ^^       "^  ,  we  shall  find 

sin  ^  4-  sin  5  =  2  sin  ^  (;;  +  (^)  cos  I  {p  —  q),  »  .     [v) 

sin  p  —  sin  J  =  2  sin  ^  {p  —  q)  cos  I  (p  +  ^),  .  .    (x) 

cos  ;:)  +  cos  7  =  2  cos  I  (p  +  7)  cos  l  {p  —  q\  -  .    (y) 

cos  2-  —  cos  ^9  =  2  sin  ^  {p  +  (?)  sin  -J  {j)  —  2),  .  (2) 


PLANE    TEIGONOMETRY.  813 

If  we  make  q  =  0^  we  shall  have, 

sill  p  =^  2  sin  \p  cos  \Pt  ,     .     .     .  (x  1) 

1  +  cos  p  =  2  cos-  Ip^      ....  (y  1) 

1  —  cos  7;  =  2  sin-  \p^      ....  (2  1) 

86.  From  formulas  (v),  (x),  (_?/),  (2),  and  {h  1),  we  obtain ; 

sm  ^)  +  sin  5-  _  sin  \  {p  +  ^)  cos  J  (/>  —  ^^)  _  tang  ?  (y)  -f  7) 
sin  p  —  sin  5'        cos  I  {p  +  (/)  sin  I  {p  —  q)  ~  tang  \  {p  —  (^j 

sin  »  +  sin  q       sin  Hp  +  ?)        .         1  /       ,      x 

1 L  — :  ^     ,         =  tans:  A  (»  +  7). 

cos  p  +  cos  ^       cos  \[p  +  q) 

sin  79  +  sin  q       cos  \{p  —  q)  ^  ,  .  . 

^ ^  =  -^ — y-^^ \  =  cot  i  (p  —  o). 

cos  ^^  —  cos  2^        sin  J  {p  —  (j)  *"  ^^         ^' 

sin  »  —  sin  q       sin  1  (  «  —  g)        ^         ,  ,  . 

cos  2^  +  cos  q  cos  2  \P  ~  q) 

smj.^_sir^  =  5^  J  Ol+_?>  =  cot  I  (p  +  q) . 

cos  q  —  COS  2^       Sin  g  (i^  +  7) 

cos  p  +  COS  7  _  COS  I  (/;  4-  q)  cos  K/^  ~"  7)  _  ^ot  I  (p  +  7) 
COS  q  —  cos/>  ~  sin  I  {p  +  (j)  sin  ^  (y:»  —  r^')  ""  tang  |  (^;  —  7)' 

sin  ^  +  sin  (7  _  2sin  J-  {p  +  (7)  cos  ^  (i?  —  7)  _  cos  lip  —  q) 
sin  (^j)  +  (?)     ~  2sin  ^  (/:>  +  g)  cos  ^  (7^  -f  7)  ~  cos  ^  (y)  +  q)' 

sin  /7  —  sin  q  __  2sin  \{p  —  q)  cos  ?,  (/7  +  ^7)        sin  \  (p  —  q) 
sin  {p  +  (/)     ~  2sin  J  (yj  +  </)  cos  |  (7;  +  (y)  ~  sin  J  (yj  +  (/)' 

These  formulas  are  the  algebraic  enunciations  of  so 
many  theorems.  The  first  expresses  that,  the  sinn  of  the 
sines  of  two  arcs  is  to  the  difference  of  those  si)ies^  as  the  tan- 
gent of  half  the  sum  of  the  arcs  is  to  the  tangent  of  half  their 
difference. 

HOMOGENEITY    OF    TERMS. 

87.  An  expression  is  said  to  be  homogeneous,  when 
each  of  its  terms  contains  the  same  number  of  literal  fac- 
tors.    Thus, 

sin-  a  +  cos-  a  =  B'      ....     (1) 

is  homogeneous,  since  each  term  contains  two  literal  flictora. 


su 


AXALYTICAL 


If  we  suppose  i?  =  1,  we  have, 

sill-  a  +  cos-  a  —  1 (2) 

This  equation  merel}-  expresses  the  numerical  relation 
between  the  values  of  sin-  a,  cos-  a,  and  unity.  If  we  pass 
from  the  radius  1  to  any  other  radius,  as  7^,  it  becomes 
necessary  to  replace  these  abstract  numbers  by  their  corres- 
ponding literal  factors.  For  this,  we  must  observe,  that 
the  radius  of  a  circle  bears  the  same 
ratio  to  any  one  of  the  functions  of  an 
arc,  (the  sine  for  example,)  as  the  radius 
of  any  other  circle,  to  the  corresponding 
function  of  a  simUar  arc  in  that  circle. 
Fur  example, 

sin  a     :  '.     R 


1 


sm  a 


hence. 


sm  a 


sm  a 


1  R 

in  which  the  sin  a.  in  the  first  member,  is  calculated  to  the 
radius  1,  and  in  the  second,  to  the  radius  R. 

If,  now,  we  substitute  this  value  of  sin  a  to  radius  1, 
in  equation  (2),  we  have, 

sin  a       sin  a       cos  a       cos  ct  _  ^ 

~~R~ ^  ~ir ^  ~ir  ^  "TT  ~   ' 

or,  sin-  a  +  cos-  a  =  R-, 

an  expression  which  is  homogeneous :  and  any  expression 
piay  be  njade  homogeneous  in  the  same  manner ;  or,  it 
may  be  made  so,  hi/  simply  multiphjirig  each  term  hy  such  a 
poicer  of  R  as  shall  yive  the  same  number  of  linear  factors  in 
all  the  terms. 

88.  Since  the  sine  of  an  arc  divided  by  the  radius  is 
equal  to  the  sine  of  another  arc  containing  an  equal  num- 
ber of  degrees  divided  by  its  radius,  we  may,  if  we  please-, 
define  the  sine  of  an  arc  to  be  the  ratio  of  the  radius  to 
the  perpendicular  let  fall  from  one  extremity  of  the  arc 
on  a  diameter  passing  through  the  other  extremity.  Giving 
similar  definitions  to  the  other  functions  of  the  arc,  each 
will  have  a  corresponding  function  in  either  angle  of  a  tri- 
angle.    For,  if  in  a  right  angled  triangle,  we  let 


PLANE   TRIGONOMETRY 


315 


A  =  right  angle;    i>' =  angle   at   base;   'C  =  vertical  angle; 

a  —  liypotlienuse ;  c  =  base ;  h  =  perpendicular, 
we  may  deduce  all  the  functions  of  the  angle  without  any 
reference  to  the  circle. 

For,  let  us  call,  bv  definition, 


sin  B  =  — )    cos  B 


a 


tan  B  =  — J    cot  B  =  -j-i 

c  b 

sec  B  =  —,  cosec  B  =  -p- 
c  0 

Each  of  these  expressions,  regarded  as  a  ratio,  is  a  mere 
abstract  number.  If  we  make  the  hypothenuse  ct  =  1,  the 
abstract  numbers  will  then  represent  pai'ts  of  a  right- 
angled  triangle,  or  the  corresponding  functions  of  a  circle 
whose  radius  is  unity. 


Formulas  rtlaiinrj  to  TrauKjles. 

89.  Let  A  CB  be  an}^  triangle,  an(J. 
designate  the  sides  b}^  the  letters  a,  6,  c ; 
then  (Art.  21), 

sin  A       a  ,  sin  A 
ein  B        b      sin  C 


sin  7>  _    b  ^    , 


a 

c      sin  C 


that  is,  tlte  sines  of  the  aivjles  are  to  each  other  as  their  oppo- 
nte  sides. 

90.  We  also  have  (Art.  22), 

a  +  b     :     a  -  b     :  :     tan  J  {A  +  B)     :     tan  \  {A  -  B)  : 

that  is,  tJie  sum  of  any  two  sides  is  to  their  difference^  as  the 
taufjerU  of  huff  the  sum  of  the  oj)p)Osite  ainjlcs  to  tlte  tan'jeni  of 
half  tlidr  difference. 

91.  In    case    of   a   ri^^ht-an^^^led    triano^le,   in    which    the 
right  angle  is  J5,  we  have  (Art.  24), 

I     :     tan  A     \  -.     c     :     a  \    hence,  a  =  c  tan  ^1,  .  (2) 

And  again  (Art.  25). 

1     :     cos  A     :  :     b    :     c  ;   hence,  c  =  b  cos  A,    .  (3) 


816  ANALYTICAT 

92.    There  is  but  one  additional  case,  that  in  which  tho 
three  sides  are  given  to  find  an  angle. 

Let  ACB  be   any  triangle,   and   CD  C 

a   perpendicular  upon  the  base.     Then,  xi\\ 

whether  the  perpendicular  falls  without  y^  J  ;    ^ 

or  within    the    triangle,  we  shall    have  y^          /    |       ^, 

(B.  IV.,  P.  12),           "  ^        c    B  D         B 

(JIj''=c=AC'  +  iB'  -  2AB  X  AD. 
But,  AD  =  AC  cos  A  \ 

and  representing    the    sides   b}'  letters,  and  substituting  for 
AD^  its  value,  \<q  have, 

70.0  o 

b~  +  c-  —  a- 
cos  A  =  ^, • 

2  be 

If  we   novr    substitute    for   cos  A^  its    value    from    formula 
(Art.  81),  we  shall  have, 

7  •">        1  o  o 

o  •   o   1   .         1         0-  +  c  -  a- 
2sm-  ^^  =  1 ^^ > 

2hc  -  (//^  +  c-  -  (r) 
=  --, ? 

Zoo 

__.  fl-^  -  7/^  -  c-  +  2hc  _  crj-  (h  -  cf 
27.c"  ~  2bc  ' 

_  (^  +  ^  -  0  (g  -{-  c  -  h) 
"  2bc 


sin  lA^x/^^^'-"^^^'^'-  'I 

If  now,  we  make 
I  {a  +  h  -\-  c)  =  s,  we  have  a  +  h  -\-  c  =  2-<,  and 
a  -[-  b  -  c  =  2.S  -  2c;   also,  a -h  c  -  b  =  2s  ~  2b : 

hence,  sin  lA  =  J k^)  ^L:il\ 

^  be 

93.    If    we    add    1    to    each    member   of    the     equation 

above,  in    Avhich    we    have    the    value    of  cos  A^  we    shall 

have, 

,        2/>  +  /.-  +  c-  -<("        {b  +  cf  -  a^ 
1  +  COS  ^  =  _  =    ^^^^ 


PLANE    TRIGONOMETRY.  317 

=  (tJ:l±J!)  Q>^<^-<')  .    and, 
2k- 

2.S  (5  -  a) 
1  +  cos  A  =  —^j • 

Substltnting  for  1  +  cos  A,  its  value   (Art.  82),   and  rcduc 
ing,  we  have, 


T    A           /  s  is  —  a 
cos  I  A  =  V -^  


9-L    Tf,    now,  we  recollect  that   the   tangent   is  equal  to 
the   sine  divided  by  the  cosine  (Art.  47),  we  have. 


^  V  g  (^s   -^    (_{^ 

and    observing   that   the   same    formula    applies   equally  to 
either  of  the  other  angles  we  have, 


^^^IB=^^^^^\^. 


tanJ(7-V-T(7:r^) — 


CONSTRUCTION   OF  TRIGONOMETRICAL   TABLES. 

95.  If  the  radius  of  a  circle  is  taken  equal  to  1,  and 
the  lengths  of  the  lines  representing  the  sines,  cosines, 
tangents,  cotangents,  &c.,  for  every  minute  of  the  quadi-ant 
be  calculated,  and  written  in  a  table,  this  would  be  a  table 
of  natural  sines,  cosines,  kc. 

96.  If  such  a  table  Avere  known,  it  would  be  easy  to 
calcuhite  a  table  of  sines,  &c.,  to  any  other  radius;  since, 
in  dilferent  circles,  the  sines,  cosines,  &c.,  of  arcs  contain- 
ing the  same  number  of  degrees,  are  to  each  other  as  their 
radii  (Art.  87). 

97.  Let  us  glance  for  a  moment  at  some  of  the  methods 
of  calculating  a  table  of  natural   sines. 

When   the    radius   of  a  circle  is  1,  the   semi-circumfer- 


818  ANALYTICAL 

ence  is  known  to  be  3.U159265358979.  This  being  divid 
ed  successively,  by  180  and  60,  or  at  once  by  10800,  gives 
.0002908882086657,  for  the  arc  of  1  minute.  Of  so  small 
an  arc,  the  sine,  chord,  and  arc,  differ  almost  imperceptibly 
from  each  other ;  so  that  the  first  ten  of  the  preceding 
figures,  that  is,  .0002908882  may  be  regarded  as  express- 
ing the  sine  of  1'  ;  and,  in  fact,  the  sine  given  in  the 
tables,  which  run  to  seven  places  of  decimals  is  .0002909 
Bv  Art.  46,  we  have, 


cos  =  v(l  —  sin^. 

This  gives,  in  the  present  case,  cos  1'  =  .9999999577.    Then 
we  ha\e  (Art.  84), 

2  cos  1'  X  sin  1'  -  sin  0'  -=  sin  2'  =  .0005817764, 
2  cos  1'  X  sin  2'  -  sin  1'  =  sin  8'  =  .0008726646, 
2  cos  1'  X  sin  3'  -  sin  2'  =  sin  4'  =  .0011635526, 
2  cos  1'  X  sin  4'  -  sin  3'  =  sin  5'  =  .0014544407, 
2  cos  1'  X  sin  5'  -  sin  4'  =  sin  6'  =  .0017453284, 
kc,  ic.,  kc. 

Thus  may  the  work  be  continued  to  any  extent,  the 
whole  difiiculty  consisting  in  the  multiplication  of  each 
successive  result  by  the  quantity  2  cos  1'  =  1.9999999154. 

Or,  having  found  the  sines  of  1'  and  2'  we  mav  deter* 
mine  new  formulas  applicable  to  further  computation. 

If  we  multiply  together  formulas  («)  and  {h)  (Art.  71-72), 
and  substitute  for  cos-  a,  1  —  sin-  a,  and  for  cos'-  b^  1  —  sin^  Z>, 
we  shall  obtain,  after  reducing, 

sin  (a  +  h)  sin  {a  —  h)  =  sin-  a  —  sin-  h ; 

and  hence,  sin  (a  + 1)  sin  (a  —  1)  =  (sin  a  +  sin  h)  (sin  a  —  sin  />) 

or,  sin  (a  —  I)  :  sin  a  —  sin  b  :  :  sin  a  +  sin  b  :  sin  (a  -f-  h). 

Applying  this  proportion,  we  have, 

sin  2'  +  sin  1    :  sin  3', 

sin  3'  4-  sin  1'    :  sin  4', 

sin  4'  -f  sin  1'   :  sin  5', 

sin  5'  -h  sin  1'    :  sin  6\ 
*kc.,                         &c.,                         ifeo. 


sin  1' 

:    sin  2' 

-  sin  1' 

sin  2' 

sin  8' 

-  sin  1' 

sin  3' 

:    sin  4' 

—  sin  1' 

sin  4' 

:    sin  5 

—  sin  1' 

pla:n:e  teigonometey 


319 


In   like   manner,  the   computer    might   proceed   for   the 
aines.of  degrees,  &c,,  thus: 


sin  1° 
sin  2° 
sin  8° 


sin  2°  —  sin  1° 
sin  3°  -  sin  1° 
sin  4°  -  sin  1° 


&c. 


sin  2°  +  sin  r    :  sin  3°, 

sin  3°  +  sin  V    :  sin  4°, 

sin  4°  4-  sin  1°    :  sin  5°, 
(tc. 


Ilaving  found  the  sines  and  cosines,  the  tangents,  co- 
tangents, secants,  and  cosecants,  may  be  computed  from 
them  (Table  I). 

98.  There  are  jet  other  methods  of  computation  and 
verification,  which  it  may  be  well  to  notice. 

Let  AP  be  an  arc  of  60°  :  then  the 
chord  AP  is  equal  to  the  radius  GA 
(b.  v.,  p.  4)  :  and  the  triangle  CPA  is 
equilateral.  Hence,  PJI  bisects  CA,  or 
cos  60°  =  i  it,  or  equal  to  one-half,  when 

But        cos  60°  =  sin  30°  (Art.  12) : 
hence,  sin  30°  =  i  ;    and, 

cos  30°  =  V  1  -  sin2  30°=  -J  VT' 
Then,  by  formulas  of  Articles  81,  and  82,   we  can  lind 
the  sine  and  cosine  of  15°,  7°  30',  3°  45',  &c. 

99.  If  the  arc  AP  were  45°,  the  right-angled  triangle 
CPJ/ would  be  isosceles,  and  we  should  have  CJI  =  PM  \ 
that  is, 

sin  45°  =  cos  45°. 
Hence,  sin^  a  -f  cos-  a  =  1, 

2  sin^  45°  =  1  : 


gives 
or, 

Also, 


sin  45°  =  cos  45°  =  -/J  =  i  y^. 

tan  45°  =  ""'^  t^°  =  1  =  cot  45°. 
cos  4o 


Above  45°,  the  process  of  computation  may*  be  simpli 
fied  by  means  of  the  formula  for  the  tangent  of  the  sum 
of  two  arcs  (Art.  75). 

ta„(45»  +  ^)  =  ;  +  *"'f.     ■ 

'        1  —  tan  b 


320  PLANE    TKIGOXO^EETRY. 

100.  If  the  trigonometrical  lines  themselves  were  used, 
it  would  be  uecessarv,  in  the  calcuhitions,  to  perforin  the 
operations  of  multiplication  and  division.  To  avoid  so 
tedious  a  method  of  calculation,  we  use  the  logarithms  of 
the  siues,  cosines,  kc. ;  so  that  the  tables  in  common  use 
show  the  values  of  the  logarithms  of  the  sines,  cosines, 
tangents,  cotangents,  kit.,  for  each  degree  and  mnnite  of 
the  quadrant,  calculated  to  a  given  radius.  This  radius  is 
10,000,000,000,  and  consequently,  its  logarithm  is  10. 

The  logarithms  of  the  secants  and  cosecants  are  not 
entered  in  the  tables,  being  easily  found  from  the  cosines 
and  sines.  The  secant  of  any  arc  is  equal  to  the  square 
of  radius  divided  by  the  cosine,  and  the  cosecant  to  the 
square  of  radius  divided  b}^  the  sine  (Table  I)  :  hence,  the 
logarithm  of  the  former  is  found  by  subtracting  the  loga- 
rithm of  the  cosine  from  20,  and  that  of  the  latter,  by 
tjubtracting  the  logarithm  of  the  sine  from  20 


y 


SPHERICAL  TRIGONOMETRY. 


1.  A  Spherical  Triangle  is  a  portion  of  the  surface 
of  a  sphere  included  by  the  arcs  of  three  great  circles 
(b.  IX,,  I).  1).  Hence,  every  spherical  triangle  has  six  parts ; 
three  sides  and  three  angles. 

2.  Spherical  Trigonometry  explains  the  processes  of 
determining,  b}^  calculation,  the  unknown  sides  and  angles 
of  a  spherical  triangle,  when  any  three  of  the  six  parts  are 
given.  For  these  processes,  certain  formulas  are  employed 
which  exj^ress  relations  between  the  six  parts  of  the  tri- 
angle. 

3.  Any  two  parts  of  a  spherical  triangle  are  said  to  be- 
ef the  same  species  when  they  are  both  less  or  both  greater 
than  90°  ;  and  they  are  of  different  species,  when  one  is 
less  and  the  other  greater  than  90°. 

4.  Let  ABC  be  a  spherical  trian- 
gle, and  P  the  centre  of  the  sphere. 
The     ano-les     of    the    trianole     are 

o  o 

equal  to  the  diedral  angles  included 

between  the  planes  which  determine 

its  sides ;    viz. :    the  angle  A  to  the 

angle   included   by  the  planes  PAB 

and  PAC]  the  angle  B  to  the  angle  included  by  the  planCvS 

PBO  and  PBA  ;    the  angle  C  to  the  angle  included  by  the 

planes  PCB  and  PCA  (b.  ix.,  d.  1).     If  we  regard  the  side 

PA  as  unity,  the  sides   CB,  CA,  AB,  of  the  spherical  triangle 

will  measure  the  angles  CPB,  CPA,  APB,  at  the  centre  of  tiie 

sphere.  Denote  these  sides  or  angles,  respectivelj^,  by  a,  6,  and  c 

5.  On  PA,  the  intersection  of  two  faces,  assume  auy 
point,  as   Ji,  and  in  the  planes  APB,   AFC,  draw  MN  and 

21 


C 


322  SPHERICAL    TRIG  O^^OMETRY 

MO,  both  perpendicular   to  the  com 
mon   intersection    PA  :     then,    OMiSf 
will  measure  the  angle  between  these 
planes    (B.  vi.,  D.  4),  and   hence,  will     P' 
be  equal    to  the  angle  A  of   the  tri- 

ngle.     Join  0  and  N  by  the  straight 

ine  ON. 

In  the  triangles  NPO  and  NMO,   vre  have  (Plane  Trig., 
Art.  92). 

C08  P  =  cos  a  =  2PN^P0 -.^03  J/=  cos  A  = ^MOxAlN 

and  by  reducing  to  entire  terms, 

^>PNX P0xcosa=  P]y '+  FO  - No';  2 J/O X MXx cos  ^  =  My''  +SFd'—NUL 

By  subtracting  the  second  equation  from  the  first,  we  have, 

i{PNxFOx  cos  a-MOX  J/iVcos  .4)  =  PN^—  MN^'-^PO^-MO'=  2PM\ 

and  by  dividing  both  members  by  2PN  X  PO,  we  have, 

MO       MX  ,        PM  ^  PM 

cos  a  -  p^-  X  pj^  X  cos  A  =  ^5^  X  ^^  • 

But  (Plane  Trig.,  Art.  88),  gives 

MO        .    ,   MN       .        PJ/  PM 

^  =  smh,  j^  =  sm  c,  pTy  =  cos  c,  -^^  =  cos  6  ; 

substituting  these  values,  we  have, 

cos  a  —  sin  h  sin  c  cos  A  =  cos  5  cos  c ; 

and  by  transposing, 

cos  a  =  cos  6  cos  c  +  sin  6  sin  c  cos  A. 

A  similar  equation  may  be  deduced  for  the  cosine  of  either 
of  the  other  sides:    hence, 

cos  a  =  cos  b  cos  c  +  sin  h  sin  c  cos  A, 

cos  6  =  cos  a  cos  c  +  sin  a  sin  c  cos  P,    >  (1) 

cos  c  =  cos  a  cos  Z)  +  sin  a  sin  h  cos  (7. 

That  is :  T^/te  cosme  o/  either  side  of  a  splieiical  triangle  is  equal 
io  the  2^^'oduct  of  the  cosines  of  the  tvm  other  sides  plus  the 
product  of  their  sines  into  the  cosine  of  their  included  angle. 

The  three  equations  (1)  contain  all  the  six  parts  of  the 
spheiical    ti'iangle.      If    three   of  the   six   quantities   which 


SPHERICAL    TRIGONOMETRY.  32:3 

enter  into  these  equations  be  given  or  known,  tlie  remain- 
ing three  can  be  determined  (Bourdon,  Art.  103) :  hence, 
if  three  paits  of  a  spherical  triangle  be  known,  the  other 
three  may  be  determined  from  them.  These  are  the 
primary  formulas  of  Spherical  Trigonometry.  They  require 
to  be  put  under  other  forms  to  adapt  them  to  logarithmic 
computation. 

6.  Let  the  angles  of  the  spherical  triangle,  polar  to 
ABO^  be  denoted  respectively  by  A\  B\  (7',  and  the  sides 
by  a',  h',  c'.     Then  (b.  ix.,  p.  6), 

a'  =  180°  -A,    h'  =  180°  -  B,   c'  =  180°  -  (7, 
A'  =  180°  -  a,   B'  =  180°  -  h,  C  =  180°  -  c. 

Since  equations  (1)  are  equally  applicable  to  the  polar  tri- 
angle, we  have, 

cos  a'  —  cos  h'  cos  c'  +  sin  b'  sin  c'  cos  A'  : 

substituting  for  a',  b',  d  and  A',  their  values  from  the  polar 
triangle,  we  have, 

—  cos  A  —  cos  B  cos  C  —  sin  B  sin   C  cos  a  ; 

and  changing  the  signs  of  the  terms,  we  obtain, 

cos  A  =  ^vn  B  sin  C  cos  a  —  cos  B  cos  C. 

Similar  equations  may  be  deduced  from  the  second  and 
thii'd  of  equations  (1) ;    hence, 

cos  ^  =  sin  ^  sin  C  cos  a  —  cos  B  cos  C,  1 
cos  B  =  sin  A  sin  C  cos  b  —  cos  A  cos  CJ   V  (2) 
cos  C  =  sin  A  sin  B  cos  c  —  cos  A  cos  B.   I 

That  is  :  The  cosine  of  either  angle  of  a  spherical  triangle^  is 
equal  to  the  product  of  the  sines  of  the  two  other  angles  into 
the  cosine  of  their  included  side^  mintis  the  product  of  the 
cosines  of  those  angles. 

7.  The  first  and  second  of  equations  (1)  give,  after 
transposing  the  terms, 

cos  a  —  cos  b  cos  c  =  sin  b  sin  c  cos  J., 
cos  b  —  cos  a  cos  c  =  sin  a  sin  c  cos  B ; 
by  adding,  we  have, 

C03  a  +  COS  6  —  cos  c  (cos  a  +  cos  6)  =  sin  r  (sin  b  coa  A  -{•  Bin  a  cum  B) ; 


im  SPHEEICAL    TKIGONOMETRY. 

and  by  substracting  the  second  from  the  first, 

cot  a  —  cos  6  4-  cos  c  (cos  a  —  cos  h)  =  sin  c  (sin  h  cos  x\.  —  siii  a  cos  B) ; 

these  equations  may  be  phiced  under  the  forms, 

(1  —  cos  r)  (cos  a  -\-  cos  V)  =  sin  c  (sin  h  cos  A  +  sin  a  cos  .5), 
(i  +  cos  c)  (cos  a  —  cos  h)  =  sin  c  (sin  6  cos^  —  sin  a  cos^) ; 

multiplying  these  ecjuations,  member  by  member,  we  obtain, 

(1  —  cos-  c')  (cos"  a  —  cos-  h)  =  sin-  c  (sin-  b  cos-  A  —  sin-  a  cos-  jB)  ; 

substituting  sin-  c  for  1  —  cos-  c,  1  —  sin^  yl  for  cos-  J.,  and 
1  —  sin-  B  for  cos-  ^,  and  dividing  by  sin-  c,  we  have, 

cos-  a  —  cos-  h  =  sin-  i  —  sin-  h  sin-  A  —  sin^  a  +  sin-  a  sin^i^: 

then,  since  cos-  a  —  cos-  Z>  =  sin-  b  —  sin-  a,  we  have, 

sm-  0  sin~  ^  =  sm-  a  sm-  i5; 

and,  by  extracting  the  square  root, 

sin  b  sin  A  =  sin  a  sin  ^. 

By  employing  the  first  and  third  of  equations  (1)  we  shall 
find, 

sin  c  sin  A  =  sin  a  sin  (7; 

and,  by  employing  the  second  and  third, 

sin  b  sin  C  =  sin  c  sin  ^ ;  hence, 

sin  A       sin  a  ,     _.       .      ,  .     . 

~ — 7^  =  -. — r  ;  or  sm  JJ  :  sm  A  :  :  sm  o  :  sm  a, 

sm  i>       sin  o  ' 


sm  A       sm  a  •     yy       •      a 

-. — 7,  =  -: ;  or  sm  6  :  sm  A  :  :  sm  c  :  sm  a, 

sm  C        sm  c  ' 

sin  (7       sin  c  •     r^       •     ^  .7 

-^ — TT  =  — ^ — ,  ;  or  sm  B  :  sin  C  :  :  sm  b  :  sm  c. 

sin  i)       sin  b 


(3) 


That  is :  In  every  spherical  triangle,   the  sines  of  tJie  angles  art 
to  each  other   as  the  sijies  of  their  opposite  sides. 

8.  Each  of  the  formulas  designated  (1)  involves  the 
three  sides  of  the  triangle  together  with  one  of  the  angles. 
These  formulas  are  used  to  determine  the  angles  when  the 
three   sides   are   known.     It   is   necessary,  however,  to   put 


SPHEKICAL    TlilGONoMETHY 


326 


them    under    another   form    to    adapt   them    to   logarithmic 
computation. 

Taking  the  first  equation,  we  have, 


cos  A  = 


cos  a  —  cos  b  cos  c 


sin  b  sin  c 
Adding  1  to  each  member,  we  have, 


1  +  cos  A  = 


cos  a  +  sin  b  sin 


cos  b  cos  c 


sin  b  sin  c 

But,       1  +  cos  A  =  2  cos^  ^^  (Plane  Trig.,  Ait.  85), 
and,  sin  b  sin  c  —  cos  b  cos  c  =  —  cos  {I  +  c)  (Art.  73) ; 


hence. 


2cos2i^ 


1     A     —    ^^^  ^   "~    ^^^  (^    ~^   ^) 


sin  b  sin 


„  ,  ,        sini(a  -jr  b  -\-  c)  sin  ^  (t^  4-  c  —  a)   ,  ^ 

or,  cos2  lA  =  -  -^ — : r— f-^^-^ -'  (Art.  85). 

'  '^  sin  6  sm  c  ^  ' 

Putting  s  =  a  -{-  b  +  c^  we  shall  have, 

J  s  =  2  (f*  +  ^  +  c)  and  ^  s  —  a  =  ^  (/;  +  c  —  a)  : 


hence, 


cos 


A  =\/^^^^  ^^^^  ^"^  (1^  ~'^^ 


sin  b  sin  c 


cos  4  i?  =    /sin  i(^)  sin  (Is-  5)^ 

V  eir>    /T»     em    /» 


sm  a  sm  c 


cos  ^  C 


=V^ 


sin  J  (5)  sin  {^s  —  c) 


sin  a  sin  6 


(i) 


9.  Had  we  subtracted  each  member  of  the  first  equa- 
tion in  the  last  article,  from  1,  instead  of  adding,  we  should^ 
by  making  similar  reductions,   have  found. 


sin  J  A 


_     /sin  ^  (fi  +  6  —  c)  sin  J  (a  +  c  —  b) 
^  sin  b  sin  c 


sm 


sm  a  sm  c 


y_B  =  \  /^  J  (a  +  6  —  c)  sin  J  (i  + 
sin  J  (7 


«). 


_     /sin  ^{a  +  c  —  b)  sin  J  (i  -f  c  —  fl) 
^  sin  a  sin  6 


(6) 


S26 


SPHEEICAL    TRIGONOMETRY. 


Puttinu 


s  =  a  +  h  +  c,  Ave  sliall  have, 


ks—a  =  }^\h-\-c—a),  }js—h=l{a-hc—b)j  and  }yS —c=^(a-{-b—c); 
hence. 


sm 


1  ^  _  .  /  sin  [}js  —  c)  sin  {^s 


^). 


sin  ^  sin  c 


.     1   7-,  /  Sin  ihs  —  c)  sm  (A^  —  a) 

V  sm  a  sm  c 


sm 


1  /7  _     /sin  (I5  —  b)  sin  (J5  —  a) 


sin  a  sin.i 
10.    From  equations  (4)  and  (6)  we  obtain, 


U6) 


^       sin  i(5)  sin  (is  —  «) 

tan  1  i5  =  ,/sm(i.s-c)  sing^-rQ^ 
^       sin  i(s)  sin  (55  —  Z*) 


H^ 


tan  }  67  =  yiin_(i^__::LA)_sm  (i^j-^)^ 
^       sin  i(5)  sin  Qs  —  c) 

11  T\'e  may  deduce  the  value  of  the  side  of  a  trian- 
gle in  terms  of  the  three  angles  by  applying  equations  (5), 
to  the  polar  triangle.  Thus,  if  a',  Z/',  c',  ^-1',  B',  6"',  repre- 
sent the  sides  and  angles  of  the  polar  triangle,  we  shall 
have  (b.  ix.,  p.  6), 

A  =  180°  -  a',  B  =  180°  -l\   C=  180°  -  c'; 

a  =  180°  -  A',  h  =  180°  -  B',  and  c  =  180°  -  C ; 

hence,  omitting  the  ',  since  the  equations  are  applicable  to 
any  triangle,  Ave  shall  have, 


cos  i  a 


^    /cos  I  {A  -h  B-  C)  cos  i{A  +  (T-  B) 


sin  B  sin  C 

^n.  i/>=v/^QS-nA4-^~    C)   C0S1(^+   C'^^     L(8) 


sin  A  sin  (7 


COS   \ 

Putting 


,    /cos  \{A  4-  6'-  ^)  cos  \{B  ■\-  C  -  A) 
^  sm  A  sin  i> 

aS'  =  yi  +  i?  +  C,  Ave  shall  have 


SPHERICAL    TRIGOXOMETllY'. 


327 


iS-  A  =  ^^^0+  B~  A),  ^S-  IJ=  1{A  4-  C-'b), 


and, 


!^'-  6'=K.i+i>'-  C); 


hence,      cos  I  a  =  ^c^ii^^t/)  cos  Q  ,Sf-J) 

V  cm     /■?   cm    f: 


sin  iy  sin  6^ 


cos 


1 7  _ 4  /cos  (I aS'  —  C)  cos  (i aS'  —  A) 
'^         ^  sin  Jl  sin  C 


cos  -ic 


/cos  (|/^^-~g)  cos^  (1  .V  -  ^1)^ 
*^  sin  ^  sin  ^* 


(9) 


12.  All  the  formulas  necessary  for  the  solution  of  splieri- 
cal  triangles,  may  be  deduced  from  equations  marked  (1). 
K  we  substitute  for  cos  b  iu  the  third  equation,  its  value 
taken  Irom  the  second,  and  substitute  for  cos-  a  its  value 
1  —  sin-  a,  and  then  divide  by  the  common  factor,  sin  a, 
we  shall  have, 

cos  c  sin  a  =  sin  c  cos  a  cos  B  +  sin  6  cos  C. 

.         ,  .      .         .7        sin  B  sin  c 
But  equations  (3)  give  sm  0  =  — -^ — y^ — ; 

hence,  by  substitution, 


cos  c  sin  a  =  sin  c  cos  a  cos  B  + 
Dividing  by  sin  c,  we  have, 

sm  a  =  cos  a  cos  JJ  + 


sm  c 


sin  5  cos  C  sin  c 

sin  0 

sin  ^  cos  (7 
sin  C 


But, 


cos 
sin 


=  cot  (Art.  55). 


Therefore,  cot  c  sin  a  =  cos  a  cos  J5  +  cot  C  sin  i?. 

Ilcmce  we  may  write  the  three  symmetrical  equations, 
cot  a  sin  ^  =  cos  h  cos  C  +  cot  A  sin  C, 
cot  i  sin  c  =  cos  c  cos  A  +  cot  i>  sin  A,   ^  (10) 
cot  c  sin  a  =  cos  a  cos  ^  +  cot  C  sin  i?. 

That  is  :    T/i  every    spherical    trianrjlc^  tlie   coiangent   of  one  Oj 
the   sides    into    the    sine  of  a  second  side^  is  eqmd  to  the  cosine 
of  the  second  side  into  the  cosine  of  the  included  anr/le,  plus  the 
cotangent  of  the  angle  opposite  the  first  side  into  the  sine  of  Uie 
included  angle. 


828  SPIIEEICAL    TRIGONOMETRY. 


NAPIERS    ANALOGIES. 

13.    If  from  the  first  and  third  of  equations  (1),  cos  c  be 
eliminated,  there  will  result,  after  a  little  redaction, 

cos  A  sin  c  =  cos  a  sin  b  —  cos  C  sin  a  cos  b. 

From  the  second  and  third  of  equations  (1),  we  get, 

cos  B  sin  c  =  cos  b  sin  a  —  cos  C  sin  b  cos  a. 

Hence,  by  adding   these    two    equations,  and   reducing,  we 
shall  have, 

sin  c  (cos  A  +  cos  B)  =  {1  —  cos  C)  sin  (a  +  b). 

.  sin  c        sin  a       sin  Z> 

But  smce,  —. — tv  =  -. ;  =  —. — r> '  ^ve  shall  have, 

'  sni  C      sm  A       sm  i>  ' 

sin  c  (sin  J.  +  sin  B)  =  sin  (7  (sin  a  -\-  sin  Z/), 

iind,       sin  c  (sin  J.  ~  sin  B)  =  sin  (7  (sin  a  —  sin  />). 

Dividing   these   two    equations,   successively,  by  the  preced- 
ing, member  by  member,   we  shall  have, 

sin  A  +  sin  B  sin  C  sin  a  +  sin  b 

cos  A  -f  cos  B      1  —  cos  (7        sin  (a  +  Z)) 

sin  J.  —  sin  B  sin  (7  sin  a  —  sin  Z) . 

cos  ^i  +  cos  B  ~  1  —  cos  (7        sin  (a  4-  i) 

reducing   these   by  the  formulas   (Plane  Trig.,  Arts.  85,  86), 
we  have, 


tangJ(^l  +  i?)  =  cot.lCrx--^lt 


i) 


COS  ^  (a  +  /;) 

■r.  ■■  XV      sin  I  (a  —  b) 

tang  i  (A  -  i?)  =  cot  ^  (7  X  ^-i^J^^^iJ- 

Kence,  two  sides,  a  and  Zj,  with  the  included  angle  C  being 
given,  the  two  other  angles  A  and  B  may  be  found  by  the 
proportions, 

cos  J  (a  +  />)  :  cos  -i  (a  -  /i)  :  :  cot  ^  (7  :  tang  ^{A  +  B), 
sin  i-(a  +  ^)  :  sin  ^  (a  —  5)  :  :  cot  ^  C  :  tang  ^(^  -  B). 


SPHERICAL    TEIGONOMETliy.  829 

We  may  api^ly  the  same  proportions  to  tlie  triangle,  polar 
to  ABGj  by  putting 

180°  -  A',  180"  -  B\  180°  -  a',  180°  -  h\  180°  -  d, 

mstead  of  o,  6,  A^  B,  C,  respectively ;  and  after  reducing 
and  omitting  tLe  accents,  we  shall  have, 

••-OS  J  {A  -h  B)  :  cos  I  {A  -  B)  :  :  tang  |c  :  tang  i(a  +  h\ 
sm  ^{A  +  B)  :  sin  }j{A  -  B)  :  :  tang  ic  :  tang  ^ (a  -  i); 

by  mea]3s  of  which,  when  a  side  c  and  the  two  adjacent 
angles  ^1  and  B  are  given,  we  are  enabled  to  find  the  two 
other  sides  a  and  h.  These  four  proportions  are  known  by 
the  name  of  Napier''s  Analogies. 

14.  In  the  case  in  which  there  are  given  two  sides  and 
an  angle  opposite  one  of  them,  there  will  in  general  be 
two  solutions  corresponding  to  the  two  results  in  Case  II., 
of  rectilineal  triangles.  It  is  also  plain,  that  this  ambi- 
guity will  extend  itself  to  the  corresponding  case  of  the 
polar  triangle,  that  is,  to  the  case  in  which  there  are  given 
two  angles  and  a  side  opposite  one  of  them.  In  every 
case  we  shall  avoid  all  false  solutions  by  recollecting, 

1st.  That  every  angle,  and  every  side  of  a  S2:)herical  trian- 
gle is  less  than  180°. 

2d.  Tliat  the  greater  angle  lies  ojyj^osite  the  greater  side,  and 
the  least  angle  opposite  the  least  side,  and  reciprocally. 

Napier's  circular  parts. 

15.  Besides  the  analogies  of  Napier  already  demonstrat 
ed,  that  Geometer  invented  rules  for  the  solution  of  all 
the  cases  of  right-angled  spherical  triangles. 

In  every  right-angled  spher- 
ical triangle  BAG,  there  are 
six  parts :  three  sides  and  three 
angles.  If  we  omit  the  con- 
sideration of  the  right  angle, 
which  is  always  known,  there 
are  five  remaining  parts,  two 
of  which  nuist  be  given  before 
the  others  can  be  determined. 


330 


SPEERICAL    TEIGOXOMETRY 


The  Circular  parts,  as  tliey 
are  called,  are  the  two  sides  c 
and  b,  about  the  right  angle, 
the  complements  of  the  oblique 
angles  B  and  C,  and  the  com- 
plement of  the  hypothenuse  a. 
Hence,  there  are  five  circular 
parts.  The  right  angle  A  not 
being  a  circular  part,  is  supposed  not  to  separate  the  cir- 
cular parts  c  and  5,  so  that  these  parts  are  considered  as 
lying  adjacent  to  each,  other. 

If  any  two  parts  of  the  triangle  are  given,  tlieir  cor- 
responding circular  parts  are  also  known,  and  tliese,  to- 
gether with  a  required  part,  will  niake  three  parts  under 
consideration.  Now,  these  three  parts  will  all  lie  torjetlier, 
or  one  of  them  will  he  separated  from  both  of  the  others.  For 
example,  if  B  and  c  were  given,  and  a  required,  the  three 
parts  considered  would  lie  together. 

But,  if  B  and  C  were  given,  and  b  required,  the  parts 
would  not  lie  together ;  for  B  woul'd  be  separated  from 
comp.  C  by  the  part  comp.  a,  and  from  b  by  the  part  c. 
In  either  case,  comp.i?  is  the  middle  pa7t.  Hence,  when 
there  are  three  of  the  circular  parts  under  consideration, 
tJie  middle  part  is  that  one  of  thera  to  which  both  of  the  others 
are  adjacent,  or  from  luhich  both  of  them  are  separated.  In 
the  former  case,  the  parts  are  said  to  be  adjacent,  and  in 
the   latter  case,    the  parts  are  said  to  be  opposite. 

This  being  premised,  we  are  now  to  prove  the  follow- 
ing theorems  for  the  solution  of  right-angled  spherical  tri- 
angles, which,  it  must  be  remembered,  apply  to  the  circu- 
lar parts,  as  already  defined. 

1st.  Radius  into  the  sine  of  the  middle  part  is  equal  to  the 
rectangle  of  die  tangenf^s  of  the  adjacent  parts. 

2d.  Radius  into  the  sine  of  the  middle  part  is  equal  to  the 
itctamjle  of  the  cosines  of  the  opposite  parts. 

These  theorems  are  proved  by  assuming  each  of  the  five 
circular  parts,  in  succession,  as  the  middle  part,  and  by 
taking  the  extremes  first  opposite,  and  then  adjacent 
Having  thus  fixed  the  three  parts  which  are  to  be  consid- 


SPnEEICAL    TRIGONOMETRY.  831 

c^ed,  take  that  one  of  the  general  equations  for  oblique- 
angled  triangles,  that  will  contain  the  three  correspond- 
ing parts  of  the  triangle,  together  with  the  right  angle ; 
then  make  A  =  90°,  and  after  making  the  reductions  cor- 
responding to  this  supposition,  the  resulting  e(|uaiiun  will 
prove  the  rule  for  that  particular  case. 

For  example,  let  comp.  a,  be  the  middle  part  and  tlie 
extremes  opposite.  The  equation  to  be  applied  in  this 
case  must  contain  a,  h,  c,  and  A.  The  first  of  equations 
(1)  contains  these  four  quantities : 

cos  a  =  cos  b  cos  c  +  sin  h  sin  c  cos  A, 

HA  =  90°cosA  =  0; 

hence,  cos  a  =  cos  h  cos  c  ; 

that  is,  radius,  which  is  1,  into  the  sine  of  the  middle 
part,  (which  is  the  complement  of  a,)  is  equal  to  the  rect- 
angle of  the  cosines  of  the  opposite  parts. 

Suppose,  now,  that  the  comple- 
ment of  a  were  the  middle  ])art 
and  the  other  parts  adjacent.  The 
equation  to  be  applied  must  con- 
tain the  four  quantities  a,  L\  6',  and 
A.     It  is  the  first  of  equations  (2): 

cos  A  — -•  sin  B  sin  C  cos  a  —  cos  B  cos  C, 
Making  A  =  90°,  we  have, 

sin  B  sin  C  cos  a  =  cos  B  cos  C, 
or,  cos  a  =  cot  B  cot  C ; 

that  is,  radius,  which  is  1,  into  the  sine  of  the  middle 
part  is  equal  to  the  rectangle  of  the  tangent  of  tlie  com- 
plement of  i?,  into  the  tangent  of  the  complement  of  (7, 
that  is,  to  the  rectangle  of  the  tangents  of  the  adjacent 
circular  2)arts. 

Let  us  now  take  the  comp.  i?,  for  the  middle  })art  and 
the  extremes  opposite.  The  two  other  parts  under  consid- 
eration will  then  be  the  perpendicular  b  and  the  comp.  of 
the  angle  C.  The  equation  to  be  applied  must  contain  the 
four  parts  A,  B^  C,  and  b:  it  is  the  second  of  equations  (2). 
cos  B  — •  sin  A  sin  C  cos  b  —  cos  A  cos  O. 


832  SPHERICAL    TRIGONOMETRY. 

Making  A  =  90°,  we  have, 

cos  B  =  sin  C  cos  h. 

Let  comp.  B  be  still  the  middle  pare  and  the  extremes 
adjacent.  The  equation  to  be  applied  must  then  contain 
the  four  parts  a,  B,  c,  and  A.  It  is  similar  to  equa- 
tions (10)*; 

cot  a  sin  f  =  cos  c  cos  B  +  cot  A  sin  B. 

But,  if  ^  =  90°,  cot  ^  =  0 ; 

hence,  cot  a  sin  c  =  cos  c  cos  B : 

or,  cos  B  =  cot  a  tang  c. 

By  pursuing  the  same  method  of  demonstration  when  each 
circular  part  is  made  the  middle  j)art,  and  making  the 
terms  homogeneous,  when  we  change  the  radius  from  1  to 
B  (Plane  Trig.,  Art.  87),  we  obtain  the  five  following  equa- 
tions, which  embrace  all  the  cases. 


it  cos  a  =  cos  h  cos  c  =  cot  B  cot  C, 
i?cos  B  —  cos  h  sin  C  =  cot  a  tang  c, 
B  cos  C  =  cos  c  sin  B  =  cot  a  tang  /;, 
/ii  sin  h  =  sin  a  sin  B=  tang  c  cot  C, 
R  sin  c  =  sin  a  sin  (7  =  tan 2"  h  cot  i>. 


(11) 


We  see  from  these  equations  that,  if  tlie  middle  part  is 
required  ice  must  hegin  the  proportion  icith  radius ;  and  iclien 
one  of  the  extremes  is  required  ice  must  begin  the  prropjortion 
with  the  other  extreme. 

We  also  conclude,  from  the  first  of  the  equations,  that 
when  the  hypothenuse  is  less  than  90°,  the  sides  b  and  c 
are  of  the  same  species,  and  also  that  the  angles  B  and 
C  are  likewise  of  the  same  species.  AVlicn  a  is  greater 
than  90°,  the  sides  b  and  c  are  of  different  species,  and 
the  same  is  ti'ue  of  the  angles  B  and  C.  We  also  see 
from  the  last  two  equations  that  a  side  and  its  opposite 
angle  are  always  of  the  same  species. 

These  properties  are  proved  by  considering  the  algebraic 
signs  which  have  been  attributed  to  the  trigonometrical 
functions,  and  by  remembering  that  the  two  members  of 
an  equation  must  always  have  the  same  algebraic  sign. 


SPHERICAL    TRIGONOMETEY.         333 

SOLUTION     OF     RIGHT- ANGLED     SPHERICAL     TRIANGLES     BY 
LOGARITHMS. 

16.  It  is  to  be  observed,  that  wlien  any  part  of  a  tri- 
angle becomes  known  by  means  of  its  sine  only,  there  may 
be  two  values  for  this  part,  and  consequently  two  triangles 
that  will  satisfy  the  question ;  because,  the  same  sine  which 
corresponds  to  an  angle  or  an  arc,  corresponds  likewise  to 
its  supplement.  This  will  not  take  place,  when  the  un- 
known quantity  is  determined  by  means  of  its  cosine,  its 
tangent,  or  cotangent.  In  all  these  cases,  the  sign  Avill 
enable  us  to  decide  wdiether  the  part  in  question  is  less  or 
greater  than  90° ;  the  part  is  less  than  90°,  if  its  cosine, 
tangent,  or  cotangent,  has  the  sign  +  ;  it  is  greater  if  one 
of  these  quantities  has  the  sign  — . 

In  order  to  discover  the  species  of  the  required  part  of 
the  triangle,  we  shall  annex  the  minus  sign  to  the  loga- 
rithms of  all  the  elements  whose  cosines,  tangents,  or  co- 
tangents, are  negative.  Then,  by  recollecting  that  the  pro- 
duct of  the  two  extremes  has  the  same  sign  as  that  of  tiie 
means,  we  can  at  once  determine  the  sign  which  is  to  be 
given  to  the  required  element,  and  then  its  species  will  be 
known. 

It  has  already  been  observed,  that  the  tables  are  calcu- 
lated to  the  radius  i?,  w^hose  logarithm  is  10  (Plane  Trig., 
Art.  100) ;  hence,  all  expressions  involving  the  circular  func- 
tions, must  be  made  homogeneous,  to  adapt  them  to  the 
logarithmic  formulas. 

EXAMPLES. 

1.  In  the  right-angled  spherical 
triangle  BAG,  right-angled  at  A^ 
there  are  given  a  =  64:°  40'  and 
b  =  42°  12' :  required  the  remain- 
ing parts. 

First,  to  find  the  side  c. 
The  hypothenuse  a  corresponds  to  the  middle  part,  and 
the  extremes  are  opposite :    hence, 

R  cos  a  =  cos  h  cos  c,  or. 


834  SPHEEICAL    TEIGONOMETRY, 


cos 
:  :    cos 

h 

E 
a 

42°  12'         ar.  comp. 
64°  40' 

log. 

0.130296 

10.000000 

9.631326 

:     cos 

c 

54°  43'  07"  . 

. 

9.761622 

To  find  the  angle  B. 

The  side  b  is  the   middle   part   and  the  extremes  oppo 
bite :    hence, 

H  sin  h  =  cos  (comp.  a)  X  cos  (comp.  B)  =  sin  a  sin  B. 

sin     a     64°  40'         ar.  comp.         log.  0.043911 

:     sin      b     42°  12'          ....  9.827189 

:  :            B 10.000000 


:     sin     B 

48°  00'  14"  .... 

To  find  the  angle  C. 

9.871100 

The  angle 

C  is  the  middle  part  and  the 

extremes  adja- 

Lient :    hence, 

R  cos  C  =  cot  a  tang  b. 

E 

ar.  comp.         log. 

0.000000 

:     cot     a 

64°  40'          .... 

9.675237 

:  :    tang   b 

42°  12'          .... 

9.957485 

:     cos     C 

64°  34'  46"  .... 

9.632722 

2.  In  a  right-angled  triangle  BAC,  there  are  given  the 
hjpothenuse  a  =  105°  34',  and  the  angle  B  =  80°  40' :  re- 
quired the  remaining  parts. 

To  find  the  ani^le  C. 

o 

The   hypothenuse  is  the  m.iddic  part  and  the  extremes 
adjacent :    hence, 

E  cos  a  =  cot  B  cot  C. 

cot    B    80°  40'        ar.  comp.        log.  0.784220  + 

:     cos    a  105°  34'          ....  9.428717  - 

::            E       ......        ,  10.000000  + 

:     cot     C  148°  30'  54"  ....  10.212937  - 

Since  the  cotangent  of  C  is  negative,  the  angle  C  is  greater 
than  90°,  and  is  the  supplement  of  the  arc  which  would 
correspond  to  the  cotangent,  if  it  were  positive. 


SPHERICAL   TRIGONOMETRY.  835 


To  find  the  side  c. 

The  angle  B  corresponds  to  the  middle  part,   and    the 
extremes  are  adjacent:    hence, 


R  cos  B  =  cot  a  tang  c. 

cot     a  105°  34'         ar.  comp.      log. 

0.555053  - 

R 

10.000000  + 

cos   B    80"  40'         .... 

9.209992  + 

tang  c  149"  47'  36" 

9.765045  - 

To  find  the  side  h. 

The   side   h   is   the   middle   part,   and   the   extremes   are 
opposite :    hence, 


R  sin  h  -■  sin  a  sin 

B. 

R      .       ar.  comp.         log. 

, 

0.000000 

sin    a  105°  34'        .         .         . 

9.983770 

sin  B    80°  40'        .         .         . 

. 

9.994212 

sin    h     71°  54'  33" 

. 

9.977982 

OF   QUADRA^s'^TAI.   TRI,\NGLES. 

17.    A   quadrantal   spherical   triangle   is   one  which   has 
one  of  its  sides  equal  to  90°.  q 

Let  BAG  be   a  quadrantal  tri-  /\ 

angle   of  which   the  side  a  =  90°.  /      \ 

If    we   pass    to   the    corresponding 
polar  triangle,  we  shall  have 

.4'  =  180°  -a  =  90°,  B'  =  180°  -  6,    "^ 

G'  =  180°  -c,a'  =  180°  -  A,  ^--^, /" 

h'  =  180°  -B,c'  =  180°  -  C;  "        "^     -^'^ 

from  which  we  see,  that  the  polar  triangle  will  be  right- 
angled  at  A'^  and  hence,  every  case  may  be  referred  to  a 
right-angled  triangle. 

But  we  can  solve    the  quadrantal    triangle  by  means  of 
the  right-angled  triangle  in  a  maimer  still  move  simple. 


336  SPHERICAL    TRIGOXOMETRY, 

Let  tiie  side  BC  of  the  quad- 
nuital  ti'iangle  BAC,  be  equal  to 
90°;  produce  the  side  CA  till  CD 
IS  equal  to  90°,  and  conceive  the 
ai'c  of  a  great  circle  to  bti  drawn 
through  i^  and  Z>.  E 

Then   C  will  be  the   pole  of  ""^-..^^  /^ 

the    arc    BD^   and    the    angle   C  a~"—~^ — — ^L^ 

will  be  measured  by  BD  (b.  ix., 

p.  4),  and  the  angles  CBD  and  D  will  be  right  angles. 
Now  before  the  remaining  parts  of  the  quadrantal  triangle 
can  be  found,  at  least  two  parts  must  be  given  in  addition 
to  the  side  BC  =  90°  ;  in  which  case  two  parts  of  the 
rio^ht-anoled  triansfle  BDA,  toij^ether  with  the  rio^ht  ansjle, 
become  known.  Hence,  the  conditions  which  enable  us  to 
determine  one  of  these  triangles,  will  enable  us  also  to 
determiiie  the  other. 

EXAMPLES. 

1.  Li  the  quadrantal  triangle  BCA,  there  are  given 
CB  =  90°,  the  angle  C  =  42°  12Cand  the  angle  ^1  =  115°  20' ; 
required  the  remaining  parts. 

Having  produced  CA  to  I),  making  CB  =  90°,  and 
drawn  the  arc  BD,  there  will  then  be  given  in  the  right- 
angled  triangle  BAD,  the  side  a  =  C=  42°  12',  and  the 
angle  BAD  =  180°  -  BAC  =  180°  -  115°  20'  =  64°  40',  to 
find  tlie  remaining  parts. 

To  find  the  side  d. 
The  side  a  is  the  middle  part,  and  the  extremes  oppo- 
site :    hence, 

E  sin  a  =  sin  A  sin  d. 

ar.  comp.         log.  0.043911 

10.000000 

9.827189 

9.871100 

To  find  the  angle  B. 

The  angle  A  corresponds  to  the  middle  part,  and  the 
extremes  are  op])osite  :    hence, 


sin 

A 

64'  40'        ar 

R 

. 

:  sin 

a 

42°  12' 

:    sin 

d 

48°  00'  14"  . 

SPHERICAL    TRIGONOMETRY.  'Sii"/ 

R  cos  A  =  sill  B  cos  a. 

cos  a  42°  12'        ar.  comp.         log.  0.1o0296 

:  E lO.UOOOOO 

:  :   cos  A  64°  40'          ....  9.(3^1326 

;    sm  B  35°  16'  53"          .         .         .  AL^^^ 

To  find  tlie  side  h. 

The   side   ^>   is   the   middle  part,  and  the  extremes   are 
adjacent :    hence, 

E  sinh  —  cot  A  tans:  a. 


R       .         .         ar.  comp.         log. 

0.000000 

:    cot    A      64°  40'          .... 

9.675237 

:  :  tang  a      42°  12'          .... 

9.957485 

:    sin     h      25°  25'  14"  .... 

9.632722 

Hence,       (LI  -  90°  -  6  =  90°  -  25°  25'  14" 

=  64°  34'  46' 

CBA  =  90°  -  ABD  =  90°  -  35°  16'  53" 

=  54°  43'  07' 

BA  =  d 

=  48°  00'  14' 

2.  In  the  right-angled  triangle  BAC^  right-angled  at  A, 
there  are  given  a  =  115°  25',  and  c  =  60°  59' :  required 
the  remaining  parts. 

{  B  =  148°  56'  45" 

A72S.  }c=    75°  30'  33" 

(  b  =  152°  13'  50" 

8.  In  the  right-angled  spherical  triangle  BAC,  right- 
angled  at  Aj  there  are  given  c  =  116°  30'  43",  and  b  = 
29°  41'  32" :   required  the  remaining  parts. 

f  C  =  103°  52'  46" 

Arts.  -Ib=    32°  30'  22" 

(  a  =  112°  48'  58" 

4.  In  a  quadrantal  triangle,  there  are  given  the  quud- 
rantal  side  =  90°,  an  adjacent  side  =  115°  09',  and  the  in- 
cluded angle  =  115°  55'  :   required  the  remaining  parts. 

(    side,        113°  18'  19" 
Arts.  \        ,        I  117°  33'  52"    - 

(  '^"S-^"^'   i  101°  40'  07" 
22 


338  SPHERICAL    TEIGONOMETRY. 

SOLUTION    OF    OBLIQUE- ANGLED    TRIANGLES   BY   LOGARIinirS. 

18.    There  are  six  cases  wliicli    occur   in  the  solutio^i  of 
oblique-angled  spherical  triangles. 

1.  Having  given  two  sides,  and  an  angle  opposite  one 
3f  them. 

2.  Having  given  two  angles,  and  a  side  opposite  one 
of  them. 

3.  Having  given  the  three  sides  of  a  triangle,  to  fmd 
the  angles. 

4.  Having  given  the  three  angles  of  a  triangle,  to  find 
the  sides. 

5.  Having  given  two  sides  and  the  included  angle. 

6.  Having  given  two  angles  and  the  included  side. 

CASE  I. 

Given  two  sides,  and   an    angle    opposite   one   of  them,  to  find 
the  remaining  parts. 

19.   For  this  case,  we  employ  proportions  (3); 

sin  a     :     sin  5     :  :     sin  J.     :     sin  B. 

Ex.  1.  Given  the  side  a  = 
44"  13'  45",  h  =  84°  14'  29", 
and  the  angle  A  =  32°  26'  07" : 
required  the  remaining  parts. 

To  find  the  angle  B. 

sin  a  44°  13'  45"      ar.  comp.     log.  0.156437 

:     sin  b  84°  14'  29"     ....  9.997808 

:  :   sin  A  32°  26'  07"     ....  9.729445 

:     sin  B  49°  54'  38",  or  sin  B'  130°  5'  22"  9.883685 

Since  the  sine  of  an  arc  is  the  same  as  the  sine  of  its 
sapplement,  there  are  two  angles  corresponding  to  the 
logarithmic  sine  9.883685,  and  these  angles  are  supple- 
ments of  each  other.  It  does  not  follow,  however,  that 
both  of  them  will  satisfy  all  the  other  conditions  of  the 
question.  If  they  do,  there  will  be  tAvo  triangles  ACB\ 
A  CB ;    if  not,  there  will  be  but  one. 


SPnERICAL    TKIGOXOMETEY.  S39 

To  determine  the  circumstances  under  wliich  tliis  ambi- 
guity arises,  we  will  consider  the  2d  of  equations  (1) 

cos  h  =  cos  a  cos  c  +  sin  a  sin  c  cos  i?, 

from  which  we  obtain, 

^       cos  b  —  cos  a  cos  c 
cos  i>  =  : . • 

sm  a  sm  c 

Now,  if  cos  b  be  greater  than  cos  a,  we  shall  have, 

cos  b  >  cos  a  cos  c, 

or,  the  sign  of  the  second  member  of  the  equation  will 
depend  on  that  of  cos  b.  Hence,  cos  B  and  cos  b  will  have 
the  same  sign,  or  i>  and  b  will  be  of  the  same  species, 
and  there  will  be  but  one  triangle. 

But  when  cos  b  >  cos  a,  then  sin  b  <  sin  a  :    hence, 

If  the  sine  of  tJie  side  oi^posite  the  required  angle  be  less  than 
the  sine  of  the  other  given  side^  there  will  be  but  one  triangle. 
If,  however,  sin  b  >  sin  a,  the  cos  b  will  be  less  than 
cos  a,  and  it  is  plain  that  such  a  value  may  then  be  given 
to  c,  as  to  render 

cos  b  <  cos  a  cos  c, 

or,  the  sign  of  the  second  member  may  be  made  to  depend 
"on  cos  c. 

We  can  therefore  give  such  values  to  c  as  to  satisfy  the 
two  equations, 

cos  b  —  cos  a  cos  c 


+  cos  B  — 
—  cos  J5  = 


sm  a  sin  c 

cos  b  —  cos  a  cos  c 
sin  a  sin  c 


hence,  if  the  sine  of  the  side  ojyposite  the  required  angle  h 
greater  tlig^n  the  sine  of  the  other  given  side,  there  will  be  tuv 
triangles  ivhich  ivill  fulfil  the  given  conditions. 

Let  us,  however,  consider  the  triangle  ACB,  in  which 
wo  are  yet  to  find  the  base  AB  and  the  angle  C.  We  can 
find  these  parts  by  dividing  the  triangle  into  two  right- 
angled  triangles.  Draw  the  arc  CD  perpendicular  to  the 
base  AB :  then,  in  each  of  the  triangles  there  will  be  given 
the  hypothenuse  and  the  angle  at  the  base.     And  generally, 


840  SPHERICAL    TRIGONOMETRY. 

when  it  is  proposed  to  solve  an  oblique-angled  triangle  by 
means  of  the  right-angled  triangle,  we  must  so  draw  the 
perpendicular,  that  it  shall  pass  tlirough  the  extremity  of  a 
given  side^   and  lie  02:)posite  to  a  given  angle. 

To  find  the  angle   C,  in  the  triangle  ACD. 

cot    A     32°  26'  07"     ar.  comp.      log.  9.803105 

:             B 10.000000 

:  :   cos     h     84°  14'  29"  ....  _9.00I465 

:    cot  ^CT  86°  21'  06".        .        .        .  8.804570 


To  find  the  angle  C  in  the  triangle  DCB. 

cot     B    49°  54'  38"     ar.  comp.     log. 

0.074810 

R 

10.000000 

cos     a     44°  13'  45"  .... 

9.855250 

cot  nCB^r  35'  38". 

9.930060 

Hence,  ACB  =  135°  6Q'  44". 


To  find  the  side  AB. 


sin  A  32°  26'  07"     ar.  comp.     log.  0.270555 

sin  C  135°  56'  44".         .         .         .  9.842198 

sin  a  44°  13'  45"  ....  9.843563 

sin  c  115°  16'  12"  ....  9.956316 


The  arc  64°  43'  48",  which  corresponds  to  sin  c  is  not 
the  value  of  the  side  AB :  for  the  side  AB  must  be 
greater  than  b,  since  ii  lies  opposite  to  a  greater  angle. 
But  b  =  84°  14'  29" :  hence,  the  side  AB  must  be  the 
gupplement  of  64°  43'  48",  or,  115°  16'  12". 

Ex.  2.  Given  b  =  91°  03'  25",  a  =  40°  36'  37",  an<l 
A  =  35°  57'  15":  required  the  remaining  parts,  when  the 
obtuse  angle  B  is  taken.* 

(  B  =  115°  35'  41" 

Ans.  }g=    58°  30'  -57" 

(  c  =    70°  68'  52" 


SPHEEICAL    TRIGONOMETRY.  341 

CASE    II. 

Having  given  two  angles  and  a  side  ojjposite  cnie  of  tlicin^    to 
find  the  remaining  parts. 

20.    For  this  case,  we  employ  the  propoilions  (3). 
sin  A    :    sin  B  :  :  sin  a    :    sin  b. 

Ex.  1.  In  a   spherical   triangle  ABC,  _.C^ 

there  are   given   the  angle  A  =  60°  12',       .--'''    /\  "'"x^ 

^  =  58°  8',  and   the   side   n  =r  62°   42' ;  "^"V.^A .\  J>M 

to  find  the  remaining  parts.  B" "A 

To  find  the  side  h. 

ar.  con:ip.  log.  0.11-i478 
9.929050 
9.9-18715 


sin 

A    50° 

12' 

sin 

B    58° 

08' 

sin 

a     62° 

42' 

sin 

h     79° 

12' 

10",  or,  100°  47'  50"        9.992243 

We  see  here,  as  in  the  last  example,  that  there  are  two 
angles  corresponding  to  the  4th  term  of  the  proportion,  and 
these  angles  are  supplements  of  each  other,  since  they  have 
the  same  sine.  It  does  not  follow,  however,  that  both  of 
them  will  satisfy  all  the  conditions  of  the  question.  K 
they  do,  there  will  be  two  triangles ;  if  not  there  will  be 
but  one. 

To  determine  when  there  are  two  triangles,  and  also 
when  there  is  but  one,  let  us  consider  the  second  of  equa- 
tions (2), 

cos  B  =  sin  A  sin  C  cos  h  —  cos  A  cos  (7, 

,  .  ,      .  ,       cos  B  +  cos  A  cos  G 

which  skives,     cos  6  =  -. — -. — 7 — -^ • 

^  sm  A  sin  6 

Now,  if  cos  B  be  greater  than  cos  A,  we  shall  have, 

cos  B  >  cos  A  cos  C, 

and  hence,  the  sign  of  the  second  member  of  the  equa- 
tion will  depend  on  that  of  cos  B^  and  consequently  cos  h 
and  cos  B  will  have  the  same  algcbi-aic  sign,  or  h  and  B 
will  be  of  the  same  species.  But  when  cos  B  >  cos  A  the 
gin  B  <  sin  A :   hence, 

If  Hie  sine  of  the   angle   opposite    the   required  side  he  lev 


3-12  SPHEKICAL    TRIGONOMETRY. 

tlian  the  sine  of  the  other  given  angle,  there  icill  he  hut  and 
solution. 

If,  however,  sin  B  >  sin  A,  the  cos  B  will  be  less  than 
cos  A^  and  it  is  plain  that  such  a  value  may  then  be  given 
tc  cos  C[  as  to  render 

cos  B  <  cos  A  cos  (7, 

or,  the  sign  of  the  second  member  of  the  equation  may 
be  made  to  depend  on  cos  C.  We  can  therefore  give  such 
values  to  C  as  to  satisfy  the  two  equations, 

cos  B  +  cos  A  cos  G 


+  cos  h  = 
and  --  cos  h  = 


sin  A  sin  G 

cos  B  4-  cos  A  cos  G 
sin  A  sin  G 


Hence,  if  the  sine  of  tlie  angle  opjjosite  tJie  required  side 
he  greater  than  the  sine  of  the  otiier  given  angle,  there  will  he 
two  solutions. 

Let  lis  first  suppose  the  side  h  to  be  less  than  90°,  or, 
equal  to  79°  12'  10".     ' 

If,  now,  we  let  fall  from  the  angle  (7,  a  perpendicular 
on  the  base  BA,  the  triangle  wil  be  divided  into  two  right- 
angled  triangles,  in  each  of  which  there  will  be  two  parts 
known  besides  the  right  angle. 

Calculating  the  parts  by  Napier's  rules,  we  find, 

G  =  130°  54'  28" 
c  =  119°  08'  26" 

K  we  take  the  side  h  =  100°  47'  50",  we  shall  find, 

G  =  156°  15'  06" 
c  =  152°  14'  18' 

Bx.  2.  In  a  spherical  triangle  ABG,  there  ore  given 
A  =  103°  59'  57'',  B  =  46°  18'  07",  and  a  =  42°  08'  48"; 
required  the  remaining  parts. 

There  will  be  but  one  triangle,  since  sin  B  <  sin  A, 

(  6  =  30° 
Ans.\  G=  36°  07'  54" 
(  c  =  24°  03'  56" 


SPHEKICAL    TRIGONOMETRY, 


Uli 


CASE   III. 

Having  giver,  the  three  sides  of  a  spherical  triangle^  to  find  Uw 

angles. 

21.    For  this  case  we  use  equations  (4). 

cos  M  =  i?\AiL?--'--'-.^-^^^- 

^  sin  h  sin  c 

Ex.  1.  In  an  oblique-angled  spherical  triangle,  there  are 
given  a  =  56°  40',  b  =  8S°  13',  and  c  =  114°  30':  requir- 
ed  the  angles. 

li^a  +  h  +  c)  =  I  s  =  127°  11'  30", 


i{b+c 

-  ci)  =  {is 

-  a)  =  70°  31'  30". 

log  sin       Is 

127°  11' 

30''  .         .        .       9.901250 

log  sin  {^s  — 

a)    70°  31' 

30"  .         .         .       9.974413 

—  log  sin         b 

83°  13' 

ar.  comp.      0.003051 

—  log  sin        c 

114°  30' 

ar.  comp.      0.040977 

Sum 

. 

.     19.919(391 

Half  sum  =  log 

cos  1 A  24° 

15'  39"    .         .       9.959845 

Hence, 


ande  A  =  48°  31'  18' 


The  addition  of  twice  the  logarithm  of  radius,  or  20, 
to  the  numerator  of  the  quantit}^  under  the  radical,  just 
cancels  the  20  which  is  to  be  subtracted  on  account  of  the 
arithmetical  complements,  so  that  the  20,  in  both  cases^ 
may  be  omitted. 

Applying  the  same  formulas  to  the  angles  B  and  (7,  we 
find, 

B=    62°  55'  46" 
C=  125°  19'  02" 

Ex.  2.  In  a  spherical  triangle  there  are  given  a  =  40" 
18'  29",    b  =  67°  14'  28",  and   c  =  89°  47'  06"  :   required 


the  three  angles. 


[A=    34°  22'  16" 

Ans.  \b=    53°  35'  16" 

I  C=  119°  13'  32" 


844 


SPEERICAL    TRIGONOMETRY, 


CASE  ly. 

Having   gicen    the   three  angles  of  a  spherical  triangle^  to  find 
the  three  sides. 

22.    For  tliis  case  we  employ  equations  (9). 
E 


cos  ^a 


^     /cos{lS-B)  cos(^S-  C) 
V  sin  i?  sin  C 


Ex.  1.  In  a  spherical  triangle  ABC  there  are  given 
A  =  48°  80',  B=  125°  20',  and  C=  62^  54'  ;  required  the 
sides. 

i(.4  +  ^  +  C)=  \S=       118°  22' 
[\  S  -  A)     .        .      =         69°  52' 


ihS-B) 

= 

-      6°  58' 

{\S-  C) 

= 

66'  28' 

log  cos  {\S-  B)  • 

-    6°  53' 

,        , 

9.996782 

log  cos  {\S-  C) 

6d°  28' 

. 

9.753495 

—  log  sin         B 

125°  20' 

ar.  com  p. 

0.088415 

—  log  sin          C 

62^  54' 

ar.  comp. 

0.050506 

Sum    .         .         .         . 

. 

. 

19.889198 

3.SL\f  sum  =  log  cos  .V 

a        28°  19' 

'  48"     . 

9.944599 

llence,  side  a  =  56°  39'  36". 

In  a  similar  manner  we  find, 

h  =  114°  29'  58" 
c  =    83°  12'  06" 

Fjc.  2.  In  a  spherical  triangle  ABC,  there  are  given 
A  =  109°  55'  42",  B=  116°  38'  33",  and  C=  120°  43'  37"; 
required  the  three  sides. 

(a=    98°  21'  40" 

Ans.  \  h  =  109°  50'  22" 

c  =  115°  13'  26" 


CASE   V. 


Having   given    in   a  spherical  triangle,  two  sides  and  their  inr 
eluded  angle,  to  find  the  remaining  ^^arte. 

23.    For  this  case   we   employ  the  two  first  of  Napier's 
Analogies. 


SJ  EIERICAL    TKIGONOMETRY.  345 

cos  ^{a  +  h)  :  cos  i{a  -  h)  : :  cot  ^  C  :   tang  |(^  +  B), 
sin  i{a  -\-  b)  :  sin  -^-(a  —  h)  ::  cot  -^-  C  :   tang  ^(^i  —  B). 

Having  found  the  half  sum  and  the  half  difference  of 
the  angles^!  and  B,  the  angles  themselves  become  known, 
f<:>r,  the  greater  angle  is  equal  to  the  half  sum  jjlus  the 
half  difference,  and  the  lesser  is  equal  to  the  half  sum 
/ninus  th;;  half  difference. 

The  greater  angle  is  then  to  be  placed  opposite  the 
greater  side.  The  remaining  side  of  the  triangle  can  be 
found  bj  Case  II. 

Ex.  1.  In  a  spherical  triangle  ABC^  there  are  given 
a  =  68°  46'  02",  b  =  37°  10',  and  C  =  39°  23' ;  to  find 
the  remaining  parts. 

i(a  +  b)  =  52°  58'  1",  l(ci  -  b)=  15°  48'  01",  ^0=  19°  41'  30". 

cos    i(a-{-b)     52°  58'  01"     log.     ar.  comp.       0.220205 

:  cos  -Ha-/>)  15°  48'  01"  .  .  .  9.983272 
::   cot    ^0  19°  41'  30"         .         .         .        10.446253 

:    tang  K^  +  -^)  77°  22'  25"        .         .         .        10.649730 

sin     ^{a  +  b)     52°  58'  01"     log.     ar.  comp.       0.097840 

;    sin     i{(i~^)      15°  48'  01"         .         .         .  9.435023 

:  :   cot    i  0  19°  41'  30"        .         .         .        10.446253 

:    tang  |(A  -  i?)  43°  37'  21"         .         .         .         9.979116 

Hence,       A  =  77°  22'  25"  +  43°  37'  21"  =  120°  59'  47" 

B  =  77°  22'  25"  -  43°  37'  21"  =    33°^  45'  03" 

side  c =43°'  37'  37" 

Er.  2.  In  a  spherical  triangle  ABC,  there  are  given 
b  =  83°  19'  42",  c  =  23°  27'  46" ;  the  contained  angle 
A  =  20°  39'  48" :   to  find  the  remaining  parts. 

IB=  156°  30    16" 

Am.  i  C=      9°  11'  48" 

(  a  =    61°  32'  12" 

CASE  VI. 

In  a  spherical  triangle,  having   given    tiuo   angles  and   the  in- 
cluded side,  to  find  the  remaining  parts. 

24.  For  this  case,  we  employ  the  second  of  Napier's 
Analoofies. 


64.6  SPIIEEICAL    TRIGOXOMETRY 

cos  I  {A  +  B)  :  cos  ^(A  —  B)  :  :  tang  |  c  :  tang  I  {a  +  b\ 
sin  i  {A  +  B)  :  sin  ^  (A  —  B)  :  :  tang  J  c  :  tang  ^  (a  —  b). 

From  wliicli  a  and  h  are  found  as  in  the  last  case.  The 
remuining  angle  can  then  be  found  by  Case  I. 

^.r.  1.  In  a  spherical  triangle  ABC^  there  are  given 
A  =  81°  38'  20",  B  =  70°  09'  Sb",  c  =  59°  16'  23" :  to 
find  the  remaining  parts. 

l{A-\-B}  =  7d°  58'  59",  1{A-B)=d''  44'  21",  ic— 29°  38'  11'. 

cos     1  {A  +  B)   75°  53'  59"  log.  ar.  comp.     0.613287 

:     cos     ^{A-B)     5°  -M'  21"  .  .         .         9.997818 

::    tang  ^c                29°  38'  11"  .  .         .       _9.755051 

•     tang  i  (a  +  h)      6Q°  42'  52"  .  .         .       10.366156 

sin     I  (A  +  B)  75°  53'  59"  log.  ar.  comp.  0.013286 

:    sin     \{a  -  B)  5°  14'  21"  .         .         .  9.000000 

::    tang  \c  29°  3b'  11"  .         .        .  9.755051 

:    tang  \{a  -  I)  3°  21'  25"  .         .         .  8.768337 

Hence,  a  =  GC,°  42'  52"  +  3°  21'  25"  =  70°  04'  17" 
b  =  66°  42'  52"  -  3°  21'  25"  =  63°  21'  27" 
angle  C        .         .         .         .      =  64°  46'  33" 

Ex.  2.  In  a  spherical  triangle  ABC,  there  are  given 
A  =  34°  15'  03",  B  =  42°  15  13",  and  c  =  76°  35'  36''  : 
to  find  the  remaining  parts. 

(a=    40°  00'  10" 

Alls.  h=    50°  10'  30'' 

(  C=  121°  S6'  19" 


MENSURATION  OF  SURFACES. 


1.  'We  determine  tlie  area,  or  contents  of  a  surflice,  by 
finding  how  many  times  the  given  surface  contains  some 
other  sujface  which  is  assumed  as  the  unit  of  measure. 
Thus,  when  we  say  that  a  square  yard  contains  9  square 
feet,  we  should  understand  that  one  square  foot  is  taken  for 
the  unit  of  measure,  and  that  this  unit  is  contained  9  times 
m  the  square  yard. 

2.  The  most  convenient  unit  of  measure  for  a  surface, 
is  a  square  whose  side  is  the  linear  unit  in  which  the  linear 
dimensions  of  the  figure  are  estimated.  Thus,  if  the  linear 
dimensions  are  feet,  it  will  be  most  convenient  to  express 
the  area  in  square  feet;  if  the  linear  dimensions  are  yards, 
it  will  be  most  convenient  to  express  the  area  in  square 
yards,   &c. 

8.  We  have  already  seen  (b.  iv.,  p.  4,  s.  2),  that  the  term, 
rectangle  or  product  of  two  lines,  designates  the  rectangle 
constructed  on  the  lines  as  sides ;  and  that  the  numerical 
value  of  this  product  expresses  the  number  of  times  which 
the  rectangle  contains  its  unit  of  measure. 

4.    To  find  the  area  of  a  square,  a  rectangle,  or  a  parallel- 
ogram. 
Multiply  tJie  base  hij  the  altitude^  and   the  product    will   he    tlie 
area  (b.  IV.,  P.  5). 
Ex.  1.    To    find    the    area  of  a  parallelogram,    the   base 
being  12.25,  and  the  altitude  8.5.  Ans.  104.125. 

2.  Whiit  is  the  area  of  a  square  whose  side  is  204.3 
feet?  Ans.  41738.49  sq.  ft. 

8.  What  are  the  contents,  in  square  3'ards,  of  a  rectan- 
gle wbose  base  is  66.3  feet,  and  altitude  33.3  feet? 

Ans.  245.31. 


S48        MENSUEATION    OF    SURFACES. 

4.  To  find  the  area  of  a  rectangular  board,  whose 
length  is  12J  feet,  and  breadth  9  inches.     Ans.  Uf  sq.  ft. 

5.  To  find  the  number  of  square  yards  of  |;ainting  in 
a  parallelogram,  ^vhose  base  is  37  feet,  and  altitude  5  feet 
8  inches.  Ans.  21  j'g. 

5.  To  find  the  area  of  a  triangle, 

CASE    1. 

"When  the  base  and  altitude  are  given. 

Multiply    the    hd.se  hy  the    altiiuck,  and    take    half  t/ie  j'^'^oducL 

Or,    muHiply    one    of    these    dimensions    hy    half  the    other 

(b.  IV.,  r.  6). 

Ex.  1.  To  find  the  area  of  a  triangle,  whose  base  is  625, 
and  altitude  520  feet.  Ans.  162500  sq.  ft. 

2.  To  find  the  number  of  square  yards  in  a  triangle, 
whose  base  is  40,  and  altitude  80  feet.  Ans.  66f. 

8.  To  find  the  number  of  square  j'ards  in  a  triangle;, 
whose  base  is  49,  and  altitude  25^  feet.       Ans.  68.7861. 

CASE   II. 

6.  When  two  sides  and  their  included  angle  are  given. 

Add  toyetlicr  the  loyarithms  of  tjie  tu:o  sides  and  the  logarith- 
mic sine  of  their  included  anyle ;  from  tJ/is  sum  snhtract 
the  loQariihm  of  the  radius^  wJiich  is  10,  and  the  rerncHJider 
will  he  the  loyaritlim  of  double  the  area  of  the  triangle 
Find ^  from  the  table,  the  nuniber  ansicering  to  this  logarithvi, 
arid  divide  it  by  2 ;    iJie  qnotient  icill  be  the  required  area. 

Let  BA  C  be  a  triangle,  in  which 
there  are  given  BA,  BC,  and  the  in- 
cluded angle  B. 

From  the  vertex.^  draw  AD  per- 

pendicular  to  the  base  BC,  and  repre-      ^'*  ^^  ^ 

sent   the    area  of  the  triangle  by   Q.     Then  (Trig.  Tk  I.), 
B    :     sin  B    :  :     BA     :     AD ; 

,  ,  ^       BA  X  sin  B 

ncnce,  AIJ  = . 

XL 

^      BCx  AD   ^,  ^  ^. 
But,  Q  =  ^ (Art.  o) : 


MENSURATION    OF    SURFACES.        8-19 

hence,  by  substituting  for  AD  its  value,   we  liave, 

^       BCX  BA  X  sin  B            ^^       BC  X  BA  x  sin  B 
Q= 2/^ or,  26=  ^^ 

Taking  the  logarithms  of  both  members,  -we  have, 
log.  2  Q  =  log.  BC  +  log.  BA  +  log.  sin  B  -  log  R; 
the  formula  of  the  rule  as  enunciated. 

Ex.  1.  What  is  the  area  of  a  triangle  whose  sides  are, 
BC  =  125.81,  BA  =  67.65,  and  the  included  angle  B  = 
57°  25'  ? 

^  +  log.  BC  125.81  2.099715 
+  log.  BA  67.65  1.760799 
+  log.  sin  B  57°  25'  9.925626 
-  log.  B       .        .     -10. 


Then,  loir.  2  0  = 


log.  2Q 3.786140 

and  2  Q  =  6111.4,  or  Q  =  3055.7,  the  required  area. 

2.  What  is  the  area  of  a  triangle  whose  sides  are  30 
and  40,  and  their  included  angle  28°  57'  ? 

Ans.  290.427. 

3.  What  is  the  number  of  square  yards  in  a  triangle 
of  which  the  sides  are  25  feet  and  21.25  feet,  and  their 
included  angle  45°  ?  Aiis.  20.8694. 

CASE   III. 

7.   When  the  three  sides  are  known. 

1.  Add  tlie  tliree  sides  together^  and  take  half  their  sum. 

2.  From  this  half-sum  subtract  each  side  separately. 

3  Multijyly  together  the  half  sum  arul  each  of  the  three  re 
mainderSj  and  the  product  will  be  the  square  of  the  area 
of  the  triangle.  Then^  extract  the  square  root  of  this  p>ro- 
ductj  for  the  required  area. 

Or,  After  having  obtained  the  three  remainders^  add  together 
the  logarithm  of  the  half  sum  and  the  logarithms  of  the 
respective  remainders^  and  divide  their  sum  by  2  :  the  quo- 
tient will  be  the  logarithm  of  the  area. 


350        MENSURATION    OF    SURFACES, 

Let  ACB  be  a  triangle:  and  denote 
the  area  by  Q  :  then,  by  the  last  case, 
we  liave, 

Q  =  lie  X  sin  J.. 
But,  we  have  (Plane  Trig.,  Art.  78), 

sin  A  =  2  sin  ^A  cos  4- A  ; 
hence,      Q  =  he  sin  ^A  cos  ^  A. 

By  substituting  in  this  equation  the  values  of  sin  ^A,  and 
cos  ^.4,  found  in  Arts.  92  and  93,  Plane  Trigonometry,  we 
obtain, 

Q  =   V  5  (5  —  rt)  (5  —  t)   {s  —  c). 
Ex.  1.  To  find  the  area  of  a  triangle  whose  three  sides 
20,  30,  and  40. 


20        45 

45 

45  half-spm. 

80        20 

30 

40 

40        25  1st  rem. 

15  2d  rem. 

5  3d  rem. 

2)90 

45  half-sum. 

Then,  45  X  25  X  15  X  5  =  84375. 

The  square  root  of  which  is  290A737,  the  required  area. 

2.  How  many  square  yards  of  plastering  are  there  in  a 
triangle  whose  sides  are  30,  40,  and  50  feet  ?        Ans.  66|-. 

8.    To  find  the  area  of  a  trapezoid. 

Add  together  the  tv:o  parallel  sides:  then  multiply  their  sum  hy 
the   altitude  of  the    trapezoid,  and  half  the  product  will  he 
the  required  area  (b.  IV.,  P.  7). 
Ex.  \:  In   a   trapezoid    the    parallel   sides   are   750   and 

1225,  and  the  perpendicular  distance  between  them  is  1540 ; 

what  is  the  area  ?  Ans.  152075. 

2.  IIow  many  square  feet  are  contained  in  a  plank, 
whose  length  is  12  feet  6  inches,  the  breadth  at  the  greater 
end  15  inches,  and  at  the  less  end  11  inches? 

Ans.  13i|  sq.  ft. 

3.  IIow  many  square  yards  are  there  in  a  trapezoid, 
whose  parallel  sides  are  240  feet,  320  feet,  and  altitude  66 
feet?  A  is.  2053 J. 


MENSURATION    OF    SUEFACES.        851 


9.    To  find  the  area  of  a  quadrilateral. 

Join  two  of  the  angles  hy  a  dkujonal,  dividing  tlie  quadrihiteral 
into  two  triangles.  Then^  from  each  of  the  other  anrjles 
let  fall  a  ferjyendicular  on  the  diagonal:  then  multiplg  the 
diagonal  hy  half  the  sum  of  tlie  two  j^erijcndiculars^  and 
the  irroduct  ivill  he  the  area. 
Fj:.  1.    What    is   the    area  of  the 

quadrilateral    ABCD,     the     diagonal 

AC  being    42,    and    the    per|)endic- 

ulars  Ug,  Bh^  equal  to  18  and  16  feet  ? 

Ans.  714. 

2.  How  many  square  yards  of  paving  are  there  in  the 
quadrilateral  whose  diagonal  is  65  feet,  and  the  two  per- 
pendiculars let  fall  on  it  28  and  %U  feet?       Ans.  222-L. 

10.    To  find  the  area  of  an  irregular  polj^gon. 

Draw  diagonals  dividing  the  irroposed  'polygon  into  trapezoids 
and  triangles.  Then  find  the  areas  of  these  figures  sep)a- 
rately^  and  add  them  together  for  the  contenU  of  the  whole 
polygon. 

Ex.  1.  Let  it  be  required  to  deter- 
mine the  contents  of  the  polygon 
ABCDE^  having  five  sides. 

Let  us  suppose  that  we  have 
measured  the  diagonals  and  perpen- 
diculars,    and     found    ^(7=86.21, 

EC=  89.11,  Bh  =  4:,  I)d=  7.26,  Aa  =  4.18:    required  the 
area.  Aiis.  296.1292. 

11.    To    find    the    area    of   a    lonor    and    irreorular  fiofure, 
bounded  on  one  side  by  a  right  line. 

1.  Ai  the  extremities  of  the  right  line  measure  the  'perpendicu- 
lar hreadths  of  the  figure;  then  divide  the  line  into  any 
number  of  equal  parts^  and  measure  the  hreadth  at  each 
point  of  division. 

2.  Add  together  the  intermediate  hreadths  and  half  the  sum  of 
the  extreme  ones:  then  multiply  this  sum  hy  one  of  the 
equal  paints  of  the  hase  line :  the  product  will  he  the  requir- 
ed area^   very  nearly. 


852         ^[ENSURATION    OF    SURFACES. 


Let  Al'^m   be  an  irivgnlnr  figure,  ^ 

haviiii;    liji-    its    base    tlie    i-ii:lit    Hue       %~ — i^ 


AE.      Divide  AE   into    equal    parts,        ; :- ^ — ^ 

and  at    the    points  of  division  J,  i^, 

(7,  D,  and  7:.',  erect  the  pei'peudicalars  Aa^  Bh^  Cc^  Dd,  Ee.^ 
to  the  base  line  AE^  and  designate  tliem  respectively  by 
the  letters  r/,  h^  c,  cZ,  and  e. 

Then,  the  area  of  the  trapezoid  ABha  =  — - — X  AB, 

h  -\-  c 
the  area  of  the  trapezoid  BCch  =  — ^—  X  BO, 

c  +  d 
the  area  of  the  trapezoid   CBdc  =  X   CB, 

d  +  e 
and  the  area  of  the  trapezoid  BEed  =  — - —  X  BE ; 

hence,  their  sum,  or  the  area  of  the  whole  figure,  is  equal  to 

since  AB,  BC,  kc,  are  equal  to  €ach  other.  But  this  sum 
is  also  equal  to 

y-^  -\-h  -\-  c  +  d  +  -^j  X  AB, 

which  corresponds  with  the  enunciation  of  the  rule. 

Ex.  1.  The  breadths  of  an  irregular  figure  at  five  equi- 
distant places  being  8.2,  7.4,  9.2,  10.2,  and  8.6,  and  the 
length  of  the  base  40:   required  the  area. 

8.2  4)40 

8-6  10  one  of  the  equal  parts. 

2)16.8  ' 

8.4  mean  of  the  extremes. 

7.4  35.2  sum. 

9.2  10 


10.2  852 


area. 


85.2  sum. 

2.  The  length  of  an  irregular  figure  being  84,  and  the 
breadths  at  six^  equidistant  places  17.4,  20.6,  14.2,  16.5,  20.1, 
and  24.4  :    what  is  the  area  ?  Ans.  1550.64. 


MENSUKATION    OF    SUEFACES, 


853 


12.    To  find  tlie  area  of  a  regular  polygon. 

Multiply  half  the  perimeter  of  the  polijgon  hy  the  apothem^ 
or  perpendicular  let  fall  from  the  centre  on  one  of  the  sides^ 
and  the  product  luill  he  the  area  required  (b.  v.,  P.  8). 

Kemark  I. — The  following  is  the  manner  of  determining 
i\\o,  perpendicular  when  one  side  and  the  number  of  sides 
of  the  regular  polj'gon  are  known : 

First,  divide  360  degrees  by  the  number  of  sides  of 
the  polygon,  and  the  quotient  will  be  the  angle  at  the 
centre ;  that  is,  the  angle  subtended  by  one  of  the  equal 
5ides.  Divide  this  angle  by  2,  and  half  the  angle  at  the 
centre  will  then  be  known. 

ISTow,  tl;ie  line  drawn  from  the  centre  to  an  angle  of  the 
polygon,  the  perpendicular  let  fall  on  one  of  the  equal 
dides,  and  half  this  side,  form  a  right-angled  triangle,  in 
which  there  are  known  the  base,  which  is  half  the  side  of 
the  polygon,  and  the  angle  at  the  vertex.  Hence,  the  per- 
pendicular can  be  determined. 

Ex.  1.  To  find  the  area  of  a  reg- 
ular hexagon,  whose  sides  are  20  feet 
each. 

())360° 

60°  =AGB,  the  angle  at  the  centre. 
30°  =  ACD,  half  the  angle  at  centre. 


0.301030 
9.937531 
1.000000 
1.238561 


Also,  CAD  =  90°  -  ^  (7Z)  =  60° ; 
and,  ATJ  =  10. 

Then,        ?>m  ACD         .         30°,  ar.  comp 
:    sin  CAD        .         60°, 
'.'.AD        .         .         10, 
:    CD         .         17.3205 
Perimeter  =  120,  and  half  the  perimeter  =  60. 
Then,  60  X  17.3205  =  1039.23,  the  area. 

2.    What  is  the  area  of  an  octagon  whose  side  is  20? 

Ans.  1931.36886. 

Remark  II. — The    area   of    a   regular   polygon    of  any 
number  of  sides    is   easily   calculated    by  the   above   rule. 

23 


354        MENSURATION    OF    SUEFACES. 

Let  Ihe  nreas  of  the  regular  polygons  Avhose  sides  are  unity 
or  1,   be  caleulateJ  and  arningcd  in  the  followin;^ 


TABLE. 


1           NAilKS. 

SIDES. 

AREAS. 

A  AMES. 

SIDES. 

AKEAS. 

Triaii:,'lf»    . 

.     3     . 

0  48:^0127 

Octagon     . 

.    8     . 

4.8284271 

1     Square 

.     4     . 

l.UOOOIMtO 

Nnnagon    . 

.    9     . 

6.1818242 

j     Peiit-jigun  . 

.     5     . 

1.7lM)4  774 

Decagon    . 

.  10     . 

7. ('.".M  2088 

1    Hexinrnn  . 

.     6     . 

2.5'.)S(i7<i2 

Undecagon 

.  11     . 

9.8656399 

1     Heptni^nm 

.     1     .     . 

3.6889124 

Dodecngon 

.  12     . 

11.1961524     1 

Now,  since  the  areas  of  similar  polygons  are  to  each 
other  as  the  squares  of  theii-  homologous  sides  (b.  iv.,  p.  27), 
we  have, 

o 

1'     :     any  side  squared    : :    tabular  area     :     area. 
Hence,  to  find  the  area  of  any  regular  polj'gon, 

1.  S'juare  the  side  of  ihe  'pohirjoii. 

2.  Tlien  mnltiphj  that  square  hi/  ihe  tabular  area  set  opposite 
iJie  ]'>'>^U'jon  of  ihe  same  number  of  sideSj  and  the  product 
will  be  the  required  area. 

Er.  1.  What  is  the  area  of  a  regular  hexagon  whose 
side  is  20? 

20'  =  400,     tabular  area  =  2.5980762. 
Hence,  2.5980762  X  400  =  10S9.2304800,  as  before. 

2.  To  find  the  area  of  a  jientagon  ^vhose  side  is  25. 

Ans.  1075.298375. 

3.  To  find  the  area  of  a  decagon  whose  side  is  20. 

Ans.  3077.68352. 

13.  To  find  the  circumference  of  a  circle  when  the  diame- 
ter is  given,  or  the  diameter  when  the  circumference  is 
given. 

Multijilij  the  diameter  by  3.1416,  ani  the  product  icill  be  the 
circumference ;  or,  divide  ihe  circumfere^nce  by  3.1416,  and 
tloe  quotient  icill  be  the  diameter. 

It  is  shown  (b.  v.,  p.  16,  s.  1),  that  the  circumference  of 
a  circle  whose  diameter  is  1,  is  3.1415926,  or  3.1416.  But, 
since  the  circumferences  of  circles  are  to  each  other  as  their 


MENSURATION    OF    SURFACES.        355 

radii  or  diameters,  we  have,  by  calling  the  diameter  of  the 

second  circle  d, 

1     :     d    :  :     3. 141 6     :     circumference, 

hence,  d  X  3.1416  =  circumference. 

_-             ,                       ,       circumference 
Hence,  also,  d  = 3X116 

Ux.  1.  What  is  the  circumference  of  a  circle  Avhoso 
diameter  is  25  ?  Ans.  78.54. 

2.  If  the  diameter  of  the  earth  is  7921  miles,  what  in 
the  cii"cumforence  ?  Ans.  24884.6136. 

3.  AVliat  is  the  diameter  of  a  circle  whose  circumfer- 
ence is  11652.1904  ?  A?-is.    3709. 

4.  AVhat  is  the  diameter  of  a  circle  whose  circumfer- 
ence is  6850?  Aiis.  2180.41. 

14.  To  find  the  length  of  an  arc  of  a  circle  containing  any 
number  of  degrees. 

Multiphj  the  iimnher  of  degrees  in  the  given  arc  hy  0.0087266, 
and  the  irroOjiet  Inj  tlie  diameter  of  the  circle. 

Since  the  circumference  of  a  circle  whose  dinmcfter  is  1, 
is    8.1416,    it    follows,    that    if   3.1416    be   divided    by    360. 
degrees,  the  quotient    will    be   the    length  of  an    arc  of  1 

degree  :    that  is,  —.7,7^.—  =  0.0087266  =  arc   of    one    degree 

to  the  diameter  1.  This  being  multiplied  by  the  numbei 
of  degrees  in  an  arc,  the  product  will  be  the  length  of 
that  arc  in  the  circle  whose  diameter  is  1;  and  this  pro- 
duct being  then  multiplied  by  the  diameter,  the  product  is 
the  length  of  the  arc  for  any  diameter  whatever. 

Remark. — "When  the  arc  contains  degrees  aiKl  minutes, 
reduce  the  minutes  to  the  decimal  of  a  degree,  which  is 
doi  e  by  dividing  them  by  60. 

Ex.  1.  To  find  the  length  of  an  arc  of  30  degrees,  the 
diameter  bei^ig  18  feet.  Ans.  4.712364. 

2.  To  find  the  length  of  an  arc  of  12''  10'  or  12^°,  tho 
diameter  being  20  feet.  Ans.  2.123472. 

3.  What  is  the  length  of  an  arc  of  10°  15',  or  10^°,  iu 
a  circle  whose  diameter  is  68  ?  Ans.  6.082390 


'666        MENSUEATION    OF    SUEFACES. 

15.   To  find  the  area  of  a  circle. 
1.    Maltijjhj  the  circumference  hy  half  the  radius   (b.  v.,  P.  15). 
Or,  2.   MuUijjly   the   square  of  Hie  radius  hy  3.1416  (b.  v.,  p. 
16). 

Ex,  1.   To  find   the   area  of  a  circle  whose  diameter  is 
10,  and  circumference  31.410.  Ans.  78.54. 

2.  Find  the  area  of  a   circle    whose  diameter  is  7,  and 
circumference  21.9912.  Ans.  38.4846. 

3.  Ilow  many  square   yards  in  a  circle  whose  diameter 
is  U  feet?  Ans.  1.069016. 

4.  What  is  the  area  of  a  circle  whose  circumference  is 
12  feet?  Ans.  11.4591. 

16.    To  find  the  area  of  a  sector  of  a  circle. 

1.  Multiply  Hie  arc  of  the  sector  hy  half  the  radius  (b.  v.,  P. 
15,  c). 

Or.  2.  Compute  tJie  area  of  the  whole  circle:  then  say,  as  360 
degrees  is  to  the  degrees  in  the  arc  of  the  sector,  so  is  tJie 
area  of  the  vjhole  circle  to  the  area  of  the  sector. 

Ex.  1.  To  find  the  area  of  a  circular  sector  whose  arc 
contains  18  degrees,  the  diameter  of  the  ch^cle  being  3  feet. 

Ans.  0.35343. 

2.  To  find  the  area  of  a   sector   whose   arc   is  20  feet, 
the  radius  being  10.  Ans.  100. 

3.  Required  the  area  of  a  sector  whose  arc  is  147°  29', 
and  radius  25  feet.  Ans.  804.3986. 

17.   To  find  the  area  of  a  segment  of  a  circle. 

L  Eind  the  area  of  tJie  sector  having  the  same  arc,  hy  the  last 
problem. 

2.  Eind  the  area  of  the  triangle  formed  hy  the  chord  of  the 
segmend  and  Hie  kvo  radii  of  the  sector. 

S.  Then  add  these  two  together  for  the  answer  ichen  the  seg- 
ment is  greater  than  a  semicircle,  and  sid>tract  the  triajigle 
from  the  sector  when  it  is  less. 


MENSURATION    OF    SURFACES.        357 

Ex.  1.  To  find  tlie  area  of  the  seg-  ^ 

ment  ACB,   its   chord  AB   being  12, 
and  tlie  radius  EA,  10  feet.  i 

EA  10  ar.  comp.        9.000000       / 

\    AD  6        .        .       0.778151      ' 

:  :    sin  Z)       90°        .         .      10.000000 
:    smAED  36°  52'  =  36.87    9.778151 

2  F 

73.74  =  the  degTees  in  tlie  arc  ACB, 

Then,  0.0087266  X  73.74  X  20  =  12.87  =  arc  ABC  nearly. 

5 

64.35  =  SLTeaEACB. 


Again,    V EA^  -  AB'  =  VlOO  -  36  =  V^  =  S  =  ED. 
and,       6  X  8  =  48  =  the  area  of  the  triangle  EAB. 
Hence,  sect.  EACB-  EAB  =  64..3d  -  48  =  16.35  =  ACB. 

2.  Find  the  area  of  the  segment  whose  height  is  18, 
the  diameter  of  the  circle  being  50.  Ans.  636.4834. 

3.  Required  the  area  of  the  segment  whose  chord  is 
16,  the  diameter  being  20.  Ans.  44.764. 

18.  To  find  the  area  of  a  circular  ring :  that  is,  the  area 
included  between  the  circumferences  of  two  circles  which 
have  a  common  centre. 

Take  the  difference  between  the  areas  of  the  two  circles, 
Dr,  subtract  the  square  of  the  less  radius  from  the  square  of  the 
greater^  and  muUijoly  the  remainder  by  3.1416. 

For  the  area  of  the  larger  is         .         .         .         7?'^, 
and  of  the  smaller r-'r. 

Their  difference,  or  the  area  of  the  ring,  is  {R  —  iry, 

Ex.  1.  The  diameters  of  two  concentric  circles  beino;  10 
and  6,  required  the  area  of  the  ring  contained  between 
their  circumferences.  Ans.  50.2656. 

2.  What  is  the  area  of  the  ring  when  the  diameters 
of  the  circles  are  10  and  20?  Ans,  235.62. 


MENSURATION  OF   SOLIDS. 


1.  The  mensuration  of  solids  is  divided  into  two  parts: 
First.    The  mensuration  of  their  surfaces;    and, 
Stcond.    The  mensuration  of  their  sol iili ties. 

2.  AVe  have  already  seen,  that  the  unit  of  measure  for 
plane  surfaces  is  a  square  whose  side  is  the  unit  of  length 

A  curved  line  which  is  expressed  by  numbers  is  also 
referred  to  a  unit  of  length,  and  its  numerical  value  is  the 
number  of  times  which  the  line  contains  its  unit.  If  then, 
we  suppose  the  linear  unit  to  be  reduced  to  a  right  line, 
and  a  square  constructed  on  this  line,  this  square  will  be 
the  unit  of  measure  for  curved  surfaces. 

8.  The  unit  of  solidity  is  a  cube,  the  f^\ce  of  which  is 
equal  to  the  superficial  unit  in  which  the  surface  of  the 
solid  is  estimated,  and  the  edge  is  equal  to  the  linear  unit 
in  which  the  linear  dimensions  of  the  solid  are  expressed 
(b.  VII.,  p.  18,  s.  1). 

T^e  following  is  a  table  of  solid  measures: 
1T'j!8     cubic  inches  =  1  cubic  foot. 
27     cubic  feet       =  1  cubic  yard. 
4492 1  cubic  feet       =  1  cubic  rod. 

OF    POLYEDIIOXS,    Oli,    SURFACES   BOUNDED   BY    PLANES. 

4.    To  find  the  surface  of  a  right  j^rism. 

Maltiphj  the  jierimetcr  of  tlte  ha.se  hy  tJie  altitude^  ar-d  the  pro- 
duct will  be  the  convex  surface  (b.  Vll.,  }\  1).  To  this  add 
the  area  of  the  two  bases,  ichen  the  entire  surface  is  required, 

Ex.  1.  To  find  the  surface  of  a  cube,  the  length  of 
each,  side  being  20  feet.  Ans.  2400  sq.  ft. 

2.  To  find  the  whole  surface  of  a  triangular  prism, 
whose  base  is  an  equilateral  triangle,  having  each  of  its 
sides  equal  to  18  inches,  and  altitude  20  feet. 

Ans.  91.949. 


MENSUKATION    OF    SOLIDS.  85i' 

3.  Wl  at  must  be  paid  for  lining  a  rectangular  cistern 
with  lead,  at  2c/.  a  pound,  the  thickness  of  the  lead  being 
such  as  to  require  Ills,  for  each  square  foot  of  surface ; 
the  inuf  r  dimensions  of  the  cistern  being  as  follows,  viz. ; 
the  l(?i^]gth  3  Icet  2  inches,  the  breadth  2  feet  8  inches,  and 
the  depth  2  feet  6  inches/'  xins.  21.  Ss.  lO^cL 

5.    To  find  the  surface  of  a  right  pyramid. 

Multiply  the  ijcrimeier  of  the  base  hy  luilf  the  slant  lu^irjlit^  and 
the  prodaci  will  he  the  corivex  surface  (B.  VIl.,  P.  4)  :  to  ihia 
add  the  area  of  the  hase^  ivhen  the  entire  surface  is  required, 

Ex.  1.  To  find  the  convex  sui'face  of  a  right  trian- 
guhir  pj^ramid,  the  slant  height  being  20  feet,  and  each 
side  of  the  base  3  feet.  Ans.  90  sq.  ft. 

2.  What  is  the  entire  surface  of  a  right  pyramid, 
whose  slant  height  is  15  feet,  and  the  base  a  pentagon,  of 
which  each  side  is  25  feet?  Ans.  2012.798. 

6.    To   find   the   convex   surface   of  the   frustum  of  a  right 

})yraniid. 

Multiphj  the  half  sum  of  the  perimeters  of  the  two  bases  hy  the 
slant  Jteifjht  of  the  frustum,  a)ul  the  product  ivill  he  the  con- 
vex sujface  (l3.  VII.,  P.  4,  C.) 

Ex.  1.  How  many  stpiare  feet  are  there  in  the  convex 
surfiice  of  the  frustum  of  a  square  pyram.icl,  -whose  slant 
height  is  10  feet,  each  side  of  the  lower  base  3  feet  4 
inches,  and  each  side  of  the  upper  base  2  feet  2  inches? 

Ans.  110  sq.  ft. 

2.  Ariiat  is  the  convex  surface  of  the  frustum  of  an 
hcptagonal  pyramid  whose  slant  height  is  55  feet,  each  side 
of  the  lower  base  8  feet,  and  each  «iide  of  the  u]-)])er  base 
4  feet?  Ans.  231^  sq.  fL 

7.    To  find  the  solidity  of  a  prism. 

1.  Find  the  area  of  the  hase. 

2.  Multlpli/  the  area  of  the  hase   hy   the   altitude^  and  the  pro 
duct  tcill  he  the  solidity  of  the  pirism  (b.  VII.,  P.  XIV). 

Kr.  1.  What  are  the  solid  contents  of  a  cube  whose 
side  is  24  inches?  Ans.  13824. 


860  MENSURATION    OF    SOLIDS. 

2.  How  many  cubic  feet  in  a  block  of  marble,  of  Avhloh 
the  length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and 
height  or  thickness  2  feet  6  inches?  Aiis.  21  J. 

8.  Uow  many  gallons  of  water,  ale  measure,  will  a 
cistern  contain,  whose  dimensions  are  the  same  as  in  the 
Jast  example?  Ans.  1291-}. 

4.  Required  the  solidity  of  a  triangular  prism,  Avhose 
height  is  10  feet,  and  the  three  sides  of  its  triangular  base 
3,  4,  and  5  feet.  Ans.  60. 

8.    To  find  the  solidity  of  a  pyramid. 

MuUijjily  the  area  of  the  base  by  one-third  of  the  altitude^  and 
the  product  will  be  the  solidity  (b.  VII.,  P.  17). 

Ex.  1.  Required  the  solidity  of  a  square  pyramid,  each 
side  of  its  base  being  30,  and  the  altitude  25. 

Ans.  7500. 

2.  To  find  the  solidity  of  a  triangular  p3'ramid,  whose 
altitude  is  30,  and  each  side  of  the  base  3  feet. 

Ans.  38.9711. 

3.  To  find  the  solidity  of  a  triangular  pyramid,  its  alti- 
tude being  14  feet  6  inches,  and  the  three  sides  of  its  base 
5,   6,  and  7  feet.  Ans.  71.0352. 

4.  What  is  the  solidity  of  a  pentagonal  pyramid,  its 
altitude  being  12  feet,  and  each  side  of  its  base  2  feet? 

Ans.  27.5276. 
6.    What  is  the  solidity  of  an  hexagonal  pj^amid,  Avhose 
altitude  is  6.4  feet,  and  each  side  of  its  base  6  inches  ? 

Ans.  1.3S564. 

9.  To  find  the  solidity  of  the  frustum  of  a  pyramid. 

Add  together  the  areas  of  the  two  bases  of  the  frustum,  and  a 
mean  proportional  between  them,  and  then  raultiply  the  sum 
by  one-tJiird  of  the  uUilude  (r,.  vii.,  p.  18). 

Kc.  1.  To  find  the  nun^ocr  of  solid  feet  in  a  piece  of 
timber,  whose  bases  are  squares,  each  side  of  the  lower 
base  being  15  inches,  and  each  side  of  the  up])er  base 
6  inches    the  altitude  being  24  feet.  Ans.  19.5, 


MENSUEATION    OF    SOLIDS 


361 


2.  Kequirecl  the  solidity  of  a  pentagonal  frustum,  whose 
altitude  is  5  feet,  each  side  of  the  lower  base  18  inches, 
and  each  side  of  the  upper  base  6  inches. 

Ans,  9.31925. 


DEFINITIONS. 

10.  A  Wedge  is  a  solid  bound-  q  jj 
ed  by  five  planes :  viz.,  a  rectangle, 
ABCD,  called  the  base  of  the  wedge ; 
two  trapezoids  ABIIG,  DOUG,  which 
are  called  the  sides  of  the  wedge, 
and  which  intersect  each  other  in 
the  edge  Gil;  and  the  two  triangles 
GBA,  HOB,  which  are  called  the  ends  of  the  wedge. 

When  AB^  the  length  of  the  base,  is  equal  to  GH,  the 
trapezoids  ABHG,  DCHG,  become  parallelograms,  and  the 
wedge  is  then  one-half  the  parallelopipedon  described  on  the 
base  ABCJDj  and  having  the  same  altitude  with  the  wedge. 

The  altitude  of  the  wedge  is  the  perpendicular  let  fall 
from  any  point  of  the  line  GU^  on  the  base  A  BCD. 

11.  A  Rectangular  Prismoid  is  a  solid  resembling 
the  frustum  of  a  quadrangular  j^yramid.  The  upper  and 
lower  bases  are  rectangles,  having  their  corrcs])onding  sides 
parallel,  and  the  convex  surface  is  made  up  of  four  trape- 
zoids. The  altitude  of  the  prismoid  is  the  perpendicular 
distance  between  its  bases. 


TO  find  the  solidity  of  the  wedge. 

Let  L  =  AB,  the  length  of 

the   base,  I  =  Gil,    the   length 

of  the  edge,  h  =  BC,  the  breadth 

of  the  base,  h  =  PG,  the  alti- 

ude  of  the  wedge. 

Then,  1-1  =  AB-  Gil 
=  A3£ 

Suppose  AB,  the  length  of 
the  base,  to  be  equal  to   Gil.  the   lengt^i  of  the   edge,  the 
solidity  will  then    be    equal    to    luilt'    the    parallelopipedon, 


^62 


MENSCJEATiOIs  OF  SOLIDS. 


having  tlic  same  base  and  the 
same  altitude  (b.  viL,  P.  7). 
lleiicf-,  the  solidity  v/ili  be 
equal    to    ^hlli    (b.  vil,  p.  14). 

If  the  length  of  the  base  is 
greater  than  that  of  the  edge, 
let  a  seetion  MXG  be  made 
parallel  to  the  end  ECU.  The 
wedge  will  then  be  divided  into  the  triangular  prism 
BCIJ-G^  and  the  quadrangular  pyramid   G-AMSD. 

Then,  the  solidity  of  the  prism 

=  \  hid ;   the  solidity  of  the  pyramid  =  Ihh  {L  —  l); 
and  their  sum, 
^hhl  +  y.h{L  -I)  =  IhhSl-^  Ihh  2L  -  \lim  =  \hh{2L  f  L), 

If  the  length  of  the  base  is  less  than  the  length  of  the 
edge,  the  solidity  of  the  wedge  will  be  equal  to  the  differ- 
ence between  the  prism  and  pyramid,  and  we  shall  have 
for  the  solidity  of  the  wedge, 

\h]d  -  \bhil  -  L)  =  \hliZl  -  lhh2l  +  lhh2L  =  lhh{2L  +  0. 

Fx.  1.  If  the  base  of  a  wedge  is  40  by  20  feet,  ihe 
edge  oo  feet,  and  the  altitude  10  feet,  what  is  the  sohdity? 

A)is.  3883.83. 
2.    The   base  of  a  wedge   being  18  feet  by  9,  the  edge 
20  feet,  and  the  altitude  0  feet,  Avhat  is  the  solidity  ? 

A)i.s.  504 


12.  To  hnd  the  solidity  of  a  rectangular  prismoid. 

Add  [orjdlier  lite  areas  of  the  tico  bases  and  four  times  the 
area  of  a  2^<^(>'(dlel  section  at  equal  distances  from  the  bases: 
then  multiply  t/ie  sum  by  one-sixth  of  tlie  altitude. 

For,  let  L  and  B  denote  the  length 
and  breadth  of  the  lower  base,  I  and 
h  the  length  and  breadth  of  the 
upper  base,  M  and  m.  the  length 
and  breadth  of  the  section  equidis- 
tant from  the  bases,  and  A  the  alti- 
tude of  the  prismoid. 

Through    the    diagonal    edges   L 


MENSURATION    OF    SOLIDS.  363 

and  /'  let  a  i)laiie  be  passed,  and  it  will  divide  the  pris- 
moid  into  two  wedges,  having  for  bases,  the  bases  of  the 
prismoid,  and    for  edges   the   lines  L  and  /'  =  /. 

The  solidity  of  these  wedges,  and  consequently,  of  tho 
prismoid,  is 

i Bh{:iL  -h  /)  +  -;  h].{2l  +  Z)  =  i //(2 BL  +  Bl  +  2hl  +  hL) 
=  I  l>{BL  +  Bl  -\-hL-\-bl  +  BL  +  hi). 
But  since  J/  is  equall}'  distant  fiom  L  and  /,  we  have, 

2J/ -/>  +  /,  and  2ni  =  B  -\-h', 
hence,  Ufm  =  {L  +  I)  X  {IJ  +  b)  =  BL  +  Bl  +  hL  +  hi 

Substituting  4Jf/n  for  its  value  in  the  preceding  equa- 
tion, and  we  have  tor  the  solidity 

lI>{BL+bl  +  -^Jfm). 

Eemakk. — This  rule  may  be  n]i])lied  to  any  prismoid 
whatever.  Foi-,  whatever  be  the  foj'in  of  the  bases,  tliere 
may  be  inseiiljed  in  each  the  same  riumber  of  rectangles, 
and  the  number  of  these  I'ectangles  may  be  made  so  great 
that  theii'  sum  in  eaeli  base  will  diller  fj-om  that  base,  by 
less  than  any  assignable  (|uantity.  Kow,  if  on  these  rect- 
angles, rectangidar  prismoids  be  con^^trncted,  their  sum  will 
differ  fi'om  the  given  pi'ismoid  by  less  than  any  assignable 
quantit}'.     llenee,   the  rule  is  general. 

7s!/'.  1.  One  of  the  bases  of  a  I'eetangular  ]>rism()id  is 
25  feet  by  20,  the  olliei'  15  feet  by  10,  and  the  altitude 
12   feet;    I'equii-ed   the  solidity.  ^-1 /^s•.   ^JTOO. 

2.  What  is  the  s<»]idity  of  a  stick  of  hewn  tiuiber, 
whose  ends  are  oO  iuL-hes  by  27,  and  24  inches  by  18,  its 
length  being  24  feet?  An^.  102   /?, 

OF   'n]V    MEASUKES   OF   THE   THREE   ROUND   BODIES. 
13.    To  find  the  surface  of  a  cylinder. 

Multiphj  the  drcKDifirerice  of  the  hftse  hy  the  altitude^  and  'he 
2)rodnct  will  he  the  convex  surface  (}?.  VI II.,  r.  1).  To  this 
add  til",  areas  of  the  tivo  hases^  ichoi  the  eidire  surface  is 
required. 


364  MENSURATION    OF    SOLIDS. 

Ex.  1.  Wliat  is  the  convex  surface  of  a  cylinder,  the 
diameter  of  whose  base  is  20,  and  whose  altitude  is  50? 

Arts.  8141.6. 
2.    Required  the  entire  surface  of  a  cylinder,  whose  alti- 
tude is  2U  feet,  and  the  diameter  of  its  base  2  feet. 

xins.  181.U472. 

14.    To  find  the  convex  surface  of  a  cone. 

Multiply  the  circumference  of  the  base  hy  half  the  slant  height 
(b.  VIII.,  P.  3) :  to  which  add  the  area  of  the  hase^  when  the 
entire  surface  is  required. 

Ex.  1.  Ptcquired  the  convex  surface  of  a  cone,  whose 
slant  height  is  50  feet,  and  the  diameter  of  its  base  8^  feet  ? 

Ans.  667.59. 
2.   Required  the   entire   surface  of  a   cone,  Avhose.  slant 
height  is  36,  and  the  diameter  of  its  base  18  feet. 

Ans,  1272.34a 

15.    To  find  the  surface  of  a  frustum  of  a  cone. 

Multiply  the  slant  heigld  of  the  frustum  hy  half  the  sum  of  the 
circumferences  of  the  two  hases^  for  the  convex  surface  (b.  VIII., 
P.  4)  :  to  which  add  the  areas  of  the  two  haseSj  when  the  entire 
surface  is   required. 

Ex.  1.  To  find  the  convex  surface  of  the  frustum  of  a 
cone,  the  slant  height  of  the  frustum  being  122"  feet,  and  the 
circumferences  of  the  bases  8.4  feet  and  6  feet.         Ans.  90. 

2.  To  find  the  entire  surface  of  the  frustum  of  a  cone, 
the  slant  height  being  16  feet,  and  the  radii  of  the  bases 
3  feet  and  2  feet.  ^7^5.  292.1688. 

16.    To  find  the  solidity  of  a  cylinder. 

Multiply  the  area  of  the  hase  hy  the  altitiule  (b.  viil.,  P.  2). 

Ex.  1.  Required  the  solidity  of  a  cylinder  whose  alti- 
tude is  12  feet,  and  the  diameter  of  its  base  15  feet. 

Ans.  2120.58. 

2.  Required  the  solidity  of  a  cylinder  whose  altitude  ia 
20  feet,  and  the  circumference  of  whose  base  is '5  feet  6 
inches  Ans.  48.144. 


MENSUEATION    OF    SOLIDS.  865 

17.    To  find  the  solidity  of  a  cone. 

Multiply  the  area  of  the  base  by  the  altitude^  and  take  one- third 
of  the  product   (b.  VIII.,  P.  5). 

Ex.  1.  Required  tlie  solidity  of  a  cone  whose  altitude 
is  27  feet,  and  the  diameter  of  the  base  10  feet. 

Ans.  706.8G. 

2.  Eeqmred  the  solidity  of  a  cone  whose  altitudQ  is  10^ 
feet,  and  the  circumference  of  its  base  9  feet. 

Ans.  22.56. 

18.    To  find  the  solidity  of  a  frustum  of  a  cone. 

Add  together  the  areas  of  the  two  bases  and  a  mean  propor- 
tional between  them^  and  then  multipily  the  sum  by  one-third 
of  the  altitude  (b.  Vlil.,  P.  6). 

Ex.  1.  To  find  the  solidity  of  the  frustum  of  a  cone, 
the  altitude  being  18,  the  diameter  of  the  lower  base  8, 
and  that  of  the  upper  base  4.  xins.  527.7888. 

2.  What  is  the  solidity  of  the  frustum  of  a  cone,  the 
altitude  being  25,  the  circumference  of  the  lower  base  20, 
and  that  of  the  upper  base  10?  Ans.  464.216. 

8.  If  a  cask  which  is  composed  of  two  equal  conic 
frustums  joined  together  at  their  larger  bases,  have  its  bung 
diameter  28  inches,  the  head  diameter  20  inches,  and  the 
length  40  inches,  how  many  gallons  of  wine  will  it  con- 
tain, there  being  231  cubic  inches  in  a  gallon  ? 

Ans.  79.0613. 

19.   To  find  the  surface  of  a  spherical  zone. 

Multiply  the  altitude  of  the  zone  by  the  circumference  of  a  great 
circle  of  the  sp)herej  and  the  p)'^oduct  will  be  the  surface  (b. 
vni.,  p.  10,  c.  2). 

Ex,  1.  The  diameter  of  a  sphere  being  42  inches,  what 
IS  the  convex  surface  of  a  zone  whose  altitude  is  9  inches? 

Ans.  1187.5248  sq.  in. 

2.  If  the  diameter  of  a  sphere  is  12i-  feet,  what  will 
be  the  surface  of  a  zone  whose  altitude  is  2  feet? 

Ans.  78.54  sq.  ft. 


866  MEXSUPvATIOX    OF    SOLIDS. 

20.    To  find  tlie  solidity  of  a  sphere. 
1.    Maltipl.ij  Lite  surface  hij  one-lhird  of  tlie  radius  (B.  vni.,  P.  14). 
Or,   2.    Cube  the  diameter    and  rnultip^ij  tlie  numher  thus  found 
hy  \r\    that  is,  by  0.52o6  (b.  VIIL,  P.  14,  S.  8). 
Er.  1.    What  is   the   solidity  of  a  sphere  whose  diame>- 
ler  is  12?  A  us.  y04.7bU8. 

2.  A\"hat  is  the  solidity  of  the  earth,  if  the  mean  diam 
cter  be  taken  equal  to  7^18.7  niile^s  ? 

Ans.   259992792083. 

21.  To  find  the  Solidity  of  a  s})herical  segment. 

Find   the   areas  of  tlie    two    Lases^  and   muUiiily    thtir   sum    hy 
half    the   heijlit   of   the   seyment ;    to    this  product   add    the 
solidity  of  a  sphere    whose   diameter    is   equal   to   the  heiyki 
of  the  seynient  (b.  VIIL,  P.  17). 
Remakk. — When    the    segment   has   but    one  base,   the 

other  is  to  be  considered  equal  to  0  (b.  viii.,  d.  15). 

Ex.  1.    What  is  tlie  solidity  of  a  spherical  segmenf,  the 

diameter  of  the    sphere    being   40,  and    the    distances  from 

the  centre  to  the  bases,  16  and  10?  Ans.  4297.7088. 

2.  AVhat  is  the  solidity  of  a  s})herical  segment  with 
one  base,  the  diameter  of  the  sphere  being  8,  and  the  alti- 
tude of  the  segment  2  feet?  Ans.  41.888. 

3.  AVhat  is  the  solidity  of  a  s]iherical  segment  with 
one  base,  the  diameter  of  the  sphere  being  20,  and  the 
altitude  of  the  segment  9  feet?  Ans.  1781.2872. 

22.  To  find  the  surface  of  a  spherical  triangle. 

1.  Compute    the   surface  of  the   sphere   on    ichich  the  trianyle  is 
fjrnial,  and  divide  it  hy  8  ;    the  quotient  will  he  the  surface 

of  the  tri-rectanyular  trianyle, 

2.  Add  the  three  anyles  toyether ;  from  their  sum  subtract  180°, 

and  divide  the  remainder  hy  90°  :  then  multiply  the  tri- 
rectangular  trianyle  by  this  quotient,  and  the  p>^'oduct  mil 
be  the  surface  of  the  trianyle  (b.  IX.,  P.  18). 

Ex.  1.   Required  the  surfoce  of  a  triangle  described  on- 
a  sphere,  whose  diameter  is  30  feet,  the  angles  being  140^, 
92°,  and  68°.  A7is^ 4:71.24:  sq.  ft. 


MENSURATION    OF    SOLIDS. 


367 


2  Hequired  the  surface  of  a  triangle  described  on  a 
sphere  of  20  feet  diameter,  the  angles  being  120"  each. 

A) IS.  314.16  sq.  ft 

23.    To  find  the  surface  of  a  spherical  polj-gon. 

1  Find  tlie  tri-rectaivjidar  triangJe  as  hfjre. 

2  From  the  sum  of  all  the  aiifjles  take  ^ihe  product  of  two 
ri<jht  a)ifjles  hij  the  nmnhtr  of  sides  less  two.  Divide  the 
remainder  hij  90^,  a.nd  multiphj  the  tri-rectanrjidar  triangle 
htj  the  quotient:  tlie  product  will  he  the  surface  of  the  poly- 
gon  (B.  IX.,  V.  19). 

Fj:.  1.  What  is  the  surface  of  a  polj^gon  of  seven  sides, 
described  on  a  s])here  whose  diameter  is  17  feet,  the  sum 
of  the  angles  being  1080°?  Aus.  226.98. 

2.  What  is  tlie  surlace  of  a  regular  polygon  of  eight 
sides,  desci"ibed  on  a  sphere  whose  diameter  is  30,  each 
angle  of  the  polygon  being  1-10°  ?  Aus.  157.08. 


OF  THE   REGULAR   POLYEDROXS. 

24.  In  determining  the  solidities  of  the  regular  poh^e- 
drons,  it  becomes  necessary  to  kno\\^,  for  each  of  them,  the 
angle  contained  between  any  two  of  the  adjacent  faces. 
The  determination  of  this  ande  involves  the  fullowinf? 
property  of  a  regular  polygon,  viz. : 

Half  die  diagoncd  w/iich  joins  the  extremities  of  tv:o  cuJjacent 
sides  of  a  regular  pohjgon^  is  equal  to  the  side  of  die  poly- 
gon multiplied,  by  tlie  cosine  of  the  angle  which  is  obtained 
by  dividing  360°  by  twice  the  number  of  sides:  dte  radius 
beiug  equal  to  unity. 

For,  let  ABODE  be  any  regular 
polygon.  Draw  the  diagonal  A  (7,  and 
from  the  centre  F,  draw  FG  perj)en- 
dicular  to  AB.  Draw  also,  AF^  FB\ 
the  latter  Avill  be  perpendicular  to 
the  diagonal  .1  (7,  and  will  bisect  it 
at  II  (b.  III.,  p.  6,  s.) 

Let  the  number  of  sides  of  the 
polygon  be  designated  by  n :   then, 


68  MENSUEATION    OF    SOLIDS. 


60°         .     _.^        ^.^      360^ 


AFB  =  ^^^^,    and  AFG  =  CAB  -    „^^ 
But,  in  the  riglit-angled  triangle  ABff,  we  have, 

o  nr\o 

All  =  AB  cos  A  =  AB  cos  ^  (Trig.,  Th.  6). 

Remark  1. — When  the  polygon  in  question  is  the  equi- 
lateral triangle,  the  diagonal  becomes  a  side,  and  conse- 
quently, half  the  diagonal  becomes  half  a  side  of  the  tri- 
angle. 

360° 
Remark    2. — The    perpendicular     BH  =  AB    sin    — — 


25.  To  determine  the  angle  included  between  two  adja- 
cent faces  of  either  of  the  regular  polyedrons,  let  us  sup- 
pose a  plane  to  be  passed  perpendicular  to  the  axis  of  a 
polj^edral  angle,  and  through  the  vertices  of  the  polyedral 
angles  which  lie  adjacent.  This  plane  will  intersect  the 
convex  surface  of  the  polyedron  in  a  regular  polygon  ;  the 
number  of  sides  of  this  polygon  v/ill  be  equal  to  the 
number  of  planes  which  meet  at  the  vertex  of  either  of 
the  polyedral  angles,  and  each  side  will  be  a  diagonal  of 
one  of  the  equal  faces  of  the  polyedron. 

Let  B  be  the  vertex  of  a  polyedral 
angle,  CD  the  intersection  of  two  adja- 
cent faces,  and  ABC  the  section  made 
in  the  convex  surface  of  the  polyedron 
by  a  plane  perpendicular  to  the  axis 
through  B. 

Through  AB  let  a  plane  be  drawn 
perpendicular  to  CB,  produced,  if  necessary,  and  suppose 
AF^  BF,  to  be  the  lines  in  which  this  plane  intersects  the 
adjacent  faces.  Then  will  AFB  be  the  angle  included 
between  the  adjacent  faces,  and  FFB  will  be  half  tho,t 
angle  which  we  will  represent  by  ^A. 

Then,  if  we  represent  by  n  the  number  of  faces  which 
meet  at  the  vertex  of  the  solid  angle,  and  by  m  the  num- 
ber of  sides  of  each  face,  we  shall  have,  from  what  has 
already  been  shown 


MENSUEATION    OF    SOLIDS. 


369 


2/1 


and  EB  =  BG  sin 


360_° 


But, 


BF 

EB 


=  sin  FEB  =  sin  ^A,  to  tlie  radius  of  unity ; 
360° 


cos 


hence, 


sm 


A  = 


2n 


sm 


360^ 
2  m 


This   formula    gives,  for   tlie    diedral    angle   formed   bj 
any  two  adjacent  faces  of  tlie 

Tetraedron      ....  70°  31'  12" 


Hexaedron 
Octaedron 
Dodecaedron 
Icosaedron 


90° 
109°  28'  18" 
116°  33'  54" 
138°  11'  23" 


Having  thus  found  the  diedral  angle  included  betAveen 
the  adjacent  faces,  we  can  easily  calculate  the  perpendicu- 
lar let  fall  from  the  centre  of  the  polj^edron  on  one  of  its 
faces,  when  the  faces  themselves  are  known. 

The  folloAving  table  shows  the  solidities  and  surfaces  of 
the  regular  polyedrons,  when  the  edges  are  equal  to  1. 

A  TABLE   OF   REGULAR   POLYEDROXS   AVHOSE   EDGES   ARE    1. 


NAMES. 

NO.    OF  FACES. 

SURFACE. 

SULiniTY. 

Tetraedron 

4 

1.7320508 

0.117«513 

Hexaedron 

6 

6.0000000 

l.OUUOOOO 

Octaedron 

8 

3.4641016 

0.4714045 

Dodecaedron 

.      12 

.       20.64572^8 

7.6631189 

Icosaedron 

.      20      . 

8.6602540       . 

2.1816950 

26.    To  find  the  solidity  of  a  regular  polj-edron. 

1.  Multiply  the  surface  hy  one-tldrd  of  tlie  perpendicular  let  faU 
from  the  centre  on  one  of  tlie  faces^  and  the  product  luill  l)e 
the  solidity. 

Or,  2.  Multiply  tlie  cube  of  one  of  the  edges  hy  the  solidity  of 
a  similar  2)olyedron,  whose  edge  is  1. 

The   first  rule  results   from   the   division  of  the   polye- 
>iron  into  as  many  equal  pyramids  as  it  has  faces,  havin" 

24 


870  MENSUKATION    OF    SOLIDS. 

their  common  vertex  at  the  centre  of  the  polyedron.  The 
second  is  proved  b}^  considering  that  two  regular  polvedrons 
having  the  same  number  of  faces  may  be  divided  into  an 
equal  number  of  similar  pyramids,  and  that  the  sum  of 
the  pyramids  which  make  up  one  of  the  polyedr^ns  will 
i>e  to  the  sum  of  the  pyramids  which  make  up  the  other 
])olyedron,  as  a  p3^ramid  of  the  first  sum  to  a  pyramid  of 
the  second  (b.  ii.,  P.  10) ;  that  is,  as  the  cubes  of  their 
homologous  edges  (b.  vii.,  r.  20) ;  that  is,  as  the  cubes  oi 
the  edges  of  the  polyedron. 

Fx.  1.    What  is  the  solidity  of  a  tetraedron  whose  edge 
is  15?  Ans.  397.75. 

2.  What   is  the  solidity  of  a  hexaedron  w-hose  edge  is 
12?  Ans.  1728. 

3.  AVhat   is   the  solidity  of  a  octaedron    wh-^se  edge  is 
20?  Ans.  3771.236. 

4.  "What  is  the  soliditj^  of  a   dodecaedron    whose   edge 
is  25?  Ans.  119756.2328.'' 

5.  What  is  the  solidity  of  an  icosaedron  whostf  edge  is 
20?  A71S.  17453.56. 


A  TABLE 


LOGARITHMS   OF  NUMBERS 


FKOil   1   TO   10,000. 


N. 

I 

Log. 

N. 

Lo-. 

N. 

Log, 

N. 

Log. 

0-000000 

26 

1-414973 

5i 

1-707570 

76 

1-880814 

2 

o-3oio3o 

27 

i-43i364 

52 

I -716003 

77 

1-886491 

3 

0-477121 

28 

I-447I58 

53 

1.724276 

1^ 

1-892095 

4 

0-602060 

29 

1-462398 

54 

1.732394 

79 

1-897627 

5 

0-698970 

3o 

1-477I2I 

55 

I-740363 

80 

1 .903090 

6 

0-778151 

3i 

I -491362 

56 

1.748188 

81 

1-908485 

7 

0- 84-5098 

32 

i-5o5i5o 

ll 

1.755875 

82 

1-913814 

8 

0-903090 

33 

i-5i85i4 

1.763428 

83 

1-919078 

9 

0-954243 

34 

I -53 1 479 

59 

1.770852 

84 

1-924279 

10 

I  -  000000 

35 

1-544068 

60 

i-778i5i 

85 

1-929419 

II 

1-041393 

36 

i.5563o3 

61 

1.785330 

86 

1-934498 

12 

I -079181 

ll 

1-568202 

62 

1.792392 

87 

1-939019 

i3 

1 -113943 

1-579784 

63 

I -799341 

88 

1-944483 

14 

1-146128 

39 

I -591065 

64 

1.806181 

89 

1-949390 

i5 

1-176091 

40 

I -602060 

65 

1.812913 

90 

1.954243 

i6 

I -204120 

41 

I -612784 

66 

1.819544 

91 

I -939041 

n 

I -230449 
1-255273 

42 

1-623249 

tl 

1-826075 

92 

I .963788 

i8 

43 

1-633468 

1-832509 

93 

1-968483 

19 

1.27S754 

44 

1-643453 

69 

1-838849 

94 

1-973128 

20 

I -3oio3o 

45 

I-6532I3 

70 

1.845098 

95 

1.977724 

21 

I-3222I9 

46 

1-662758 

71 

1.851258 

96 

1-982271 

22 

1-342423 

47 

1-672098 

72 

1.857333 

97 

1-986772 

23 

I -361728 

48 

1-681241 

73 

1-863323 

98 

1-991226 

24 

I.3'^02II 

49 

1-690196 

74 

1-869232 

99 

I -995633 

25 

1.397940 

5o 

1-698970 

75 

1.875061 

100 

•2 -000000 

Remark.  In  the  following  table,  in  the  nine  right  hand 
columns  of  each  page,  where  the  first  or  leading  figures 
change  from  9's  to  O's,  points  or  dots  are  introduced  in- 
stead of  the  O's,  to  catch  the  eye,  and  to  indicate  that  from 
thence  the  two  figures  of  the  Logarithm  to  be  taken  from 
the  second  column,  stand  in  the  next  line  below. 


A  TABLE   OF   LOGARITHMS   FROM   1   TO    10,000. 


!  N.  ;   0 

I 

2 

3 

4 

5 

' 

7 

8 

9 

D. 

lOO  oooooo 

0434 

0868 

i3oi 

1734 

2166 

2598 

3029 

3461 

3S91 

43a 

101  ;   4321 

4751 

5i8i 

5609 
9876 

6o38 

6466 

6894 

7321 

7748 

8174  428' 

1  I02   86oo 

9026 
3259 

?^5i 

•3  00 

•724 

1 147 

.570 

1993 

2413  1  424' 

1  io3  OI2S37 

4100 

4321 

4940 

536o 

5779 

6197 

6616  4191 

1  io4   7o33 

745 1 

7868 

8284 

8-00 

91.6 

3252 

9532 
3664 

9947 

•36 1 

•775 1 416; 

io5  0211S9 

i6o3 

2016 

2428 

2841 

4073 

4486 

4896  :  412 

106  ,  53oj 

5715 

6i25 

6533 

6942 

73  5o 

7737 

8164 

8571 

8978 :  408 

107  !  93  S4 

9789 

•195 

•600 

1004 

1408 

1812 

2216 

2619 

3o2i  i  4041 

108  033424 

3o26 

4227 

46-28 

5029 

543o 

5830 

6230 

6629 

7028  1  400 

109  1  7426 

7825 

8223 

8620 

90.7 

9414 

9811 

•207 

•602 

•998 

396 

no  041393 

1787 

2182 

2576 

& 

3362 

3755 

4148 

4540 

4932 
8830 

3q3 

ni  1  53i3 

5714 

6io5 

6495 

7275 

7664 

8o53 

8442 

3^9 

112  '  9218 

9606 

9993 

•380 

•766 

ii53 

1 538 

1924 

23o9 

2694 

386 

Ii3  ;o53o-78 

3463 

3S46 

423o 

46i3 

4996 

5378 

5760 

6142 

6324 

332 

114  '  6905 

7286 

7666 

8046 

8426 

88o5 

9185 

9563 
3333 

9942 

•320 

n 

ii5  060698 

io75 

1452 

1829 

2206 

2582 

2958 

3709 

4oS3 

116  i  4458 

4832 

5206 

558o 

5953 

6326 

6699 

7071 

7443 

7815 

372 

117  1  8i86 

8557 

8928 

9298 

9668 

••38 

•407 

•776 

1145 

i5i4 

369 

iiS  071882 

2230 

2617 

29?:i5 

3352 

3718 

4o85 

445i 

4816 

5i82 

366 

119  1  5547 

5912 

6276 

6640 

7004 

7368 

773 1 

8094 

8457 

8819 

363 

120  079181 

9^43 

3i44 

9904 

•266 

•626 

•987 

1347 

1707 

2067 

2426 

36o 

121  ob2785 

35o3 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

357 

122   636o 

6716 

7071 

7426 

7781 

8i36 

8490 

8845 

9198 

9552 
3071 

355 

1 23   99o5 

•258 

•611 

•963 

i3i5 

1667 

2018 

2370 

2721 

35 1 

124  093422 

3772 

4122 

4471 

4820 

5169 

55i8 

5866 

62i5 

6562 

349 

125  1  6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

••26 

346 

126  ! 10037 1 

0715 

1059 

i4o3 

1747 

2091 

2434 

2777 

3119 

3462 

343 

127  j  3-^04 

128  1  7210 

4146 

4487 

4828 

5169 

55io 

585 1 

6191 

653 1 

6S71 

340 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

•253 

338 

129  1 10590 

0926 

1263 

1599 

1934 

2270 

26o5 

2940 

3275 

3609 

335 

i3o  j 1 13943 

4277 

4611 

4944 

5278 

56ii 

5943 

6276 

6608 

6940 

333 

i3i  1  7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

991 5 

•245 

33o 

l32  |l2o574 

0903 

I23l 

i56o 

18S8 

2216 

2544 

2811 

3198 

3523 

328 

i33  1  3852 

4178 

45o4 

4830 

5i56 

5481 

58o6 

6i3i 

6436 

6781 

325 

1  i34  j  7io5 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

••12 

323 

i35  !i3o3]i 

0655 

0977 

1298 

1619 

1939 

2260 

258o 

2900 

3219 

321 

i36  i  3539 

3858 

4177 

4496 

4814 

5.33 

545 1 

5769 

6086 

6403 

3i8 

i37  i  672. 

7037 

7354 

7671 

79'^7 

83o3 

8618 

8934 

92 '»9 

9^64 

3i5 

1 38  1  0879 

139  |i4Joi5 

•194 

•5o8 

•822 

11  36 

i45o 

1763 

2076 

2389 

2702 

3i4 

3327 

3639 

3951 

4263 

4574 

4885 

5196 

5507 

58i8 

3ii 

140  !i46i28 

6438 

674S 

7058 

7367 

7676 

7985 

B294 

86o3 

89,1 

309 

141 

9219 

9527 

9835 

•142 

•449 

•756 

io63 

1370 

1676 

19S2 

3o7 

142 

I522S.S 

2594 

2900 

32o5 

3510 

38i5 

4120 

4424 

4728 

5o32 

3o5 

143 

5336 

56  io 

5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

3o3 

144 

8362 

8664 

8965 

9266 

9567 

9868 

•168 

•469 

•765 
3738 

1068 

3oi 

145 

i6i36S 

1667 

1967 

2266 

2  564 

2863 

3i6i 

3460 

4o55 

299 

146 

43)3 

46  5o 

4947 

52  U 

5541 

5838 

6i34 

643o 

6726 

7022 

297 

"47 

7317 

7613 

7908 

82o3 

8497 

8792 

9086 

93S0 

9674 

9968 

293 

148  1702^2 

o555 

0848 

1141 

1434 

172& 

2019 

23ll 

2603 

2S95 

293 

.49   3i86 

3478 

3769 

4060 

435i 

4641 

4932 

5222 

55i2 

58o2 

291 

i5o  1 7609 1 

638i 

6670 

6959 

724S 

7536 

7825 

8ii3 

8401 

?s? 

289 

i5i   8977 
i52  181844 

9264 

9552 

gS39 

•126 

•4i3 

•699 

•985 

1272 

287 

2129 

24i5 

2700 

29S5 

3270 

3555 

3439 

4123 

4407 

285 

1 53   4691 

% 

52^9 

5542 

5825 

6108 

6391 

6674 

6956 

-239 

283 

1 54   7521 

80S4 

8366 

8647 

8928 

9209 

9490 

9771 

••5i 

281 

1 55  190332 

0612 

0892 

1171 

I45i 

i73o 

2010 

2289 

2567 

2846 

279 

id6   3i25 

34o3 

368 1 

3959 

4237 

4514 

4-92 

5069 

5346 

5623 

2nH 

1 57   5-'99 

6n6 

6453 

6^29 

7oo5 

72S. 

7556 

7832 

8107 

83S2 

276 

i5S   86D7 

8932 

9206 

9481 

9755 

••29 

•3o3 

•577 

•85o 

I12i 

274 

139  201397 

1670 

I9i3 

2216 

2488 

2761 

3o33 

33o5 

3577 

38^8 

272 

N. 

0 

2 

3 

4 

5 

6 

' 

8 

' 

D. 

A 

TABLE  OF 

LOGAIUTII 

MS  FI1 

.OM  ] 

L  TO 

10,00 

0. 

«: 

N. 

0  1  . 

2   j   3   i   4     5     6 

i  7 

;  8  1  9  1  D. 

1 60 

204120  4391 

4^)63  1  4934  1  5204 

5475 

5746 

6016 

!  6286 

6556  i  271 

161 

6826  7096 

7365 

7634  79^4 

8.73 

8441 

8710 

i388 

8979 

9247  1  269 1 

162 

951 5  97H3 

••5i 

•3 19   *j86 

•853 

1121 

1  1654 

1921  1  26/- 
4579  1  266 1 

1 63 

21.2188  2454 

2720 

2986   3252 

35i8 

3783 

1  4049 

1  43i4 

164 

4844  5 1 09 

5373 

5638  i  5902 

6166 

6430 

6694 

6957 

7221  j  264: 

i65 

74-^4  7747 

8010 

8273 

8536 

8798 

'9060 

9323 

9585 

9846  1  262; 

166  I220108  0370 

o63i 

0892 

u53 

1414 

1675 

1036 

2,96 

2456 

■   261 1 

167 

2716  2976 

3236 

3496 

3755 

4oi5 

4274 

4033 

4792 

5o5i 

i  239  i 

168 

5309  5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

2  5di 

169 

7887 

;  8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

•193 

2561 

170 

23o449 

0704 

0960 

I2l5 

1470 

1724 

'979 

2234 

2488 

2742 

254  j 

171 

2996 

3230 

3004 

3757 

4011 

4264 

4517 

4770 

5o23 

5276 

1  253 1 

172 

5328 

5781 

6o33 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

232 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

••5o 

•3oo 

1  25o 

174 

240549 

0799 

1048 

1297 

1 546 

1795 

2044 

2293 

2541 

2790 

249 

175 

3o3S 

3286 

3534 

3782 

4o3o 

4277 

4523  i  4772 

5019 

5266 

248 

176 

55i3 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

7973 

8219 

8464 

8709 

•8954 

9198 

9443 

9687 

9932 

•176 

245 

178 

200420 

0664 

0908 

ii5i 

1395 

1 638 

1881 

2125 

2368 

2610 

243 

179 

2853 

3096 

3338 

3580 

3822 

4064 

43o6 

4548 

4790 

5o3i 

242 

180 

255273 

55,4 

5755 

5996 

6237 

6477 

6718 

6958 
9355 

7198 

lit 

241 

181 

7679 

7918 

8i58 

8398 

8637 

8877 

9116 

9594 

239 

182 

260071 

o3io 

o548 

0787 

1025 

1263 

IDOI 

1739 

•976 
4346 

2214 

238 

i83 

245 1 

2688 

2925 

3162 

3399 

3636 

38i3  1  4109 

4582 

237 

184 

4S1H 

5o54 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

1 85 

7172 

7406 

7641 

7S75 

8iio 

8344 

8578 

8812 

9046 

9279 

234 

186 

95.3 

9746 

9980 

•2l3 

•446 

•679 

•912 

1144 

1377 

1609 

233 

187 

271842 

2074 

23o6 

2538 

2770 

3ooi 

3233 

3464 

3696 

3927 

232 

188 

41 58 

4389 

4620 

485o 

5o8i 

■"5311 

5542 

5772 

6002 

6232 

230 

189 

6462 

6692 

6921 

7i5i 

7380 

7609 

7838 

8067 

8296 

8525 

229 

190 

27^^754 

8982 

92 1 1 

9439 

9667 

9895 

•I23 

•35i 

•578 

•806 

228 

191 

2S1033 

1261 

1488 

,7.5 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

33oi 

3527 

3753   3970 

42o5 

443 1 

4636 

4882 

5i07 

5332 

226 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7i3o 

7354 

7578 

220 

194 

7802 

8026 

8249 

8473   8696 

8920 

9.43 

9366 

9589 
i8i3 

9812 

223 

19^ 

290035 

0257 

04S0 

0702 

0923 

1 147 

1 369 

1391 

2o34 

222 

.96 

22  56 

2478 

,2699 

2920 

3,4. 

3363 

3584 

38o4 

4025 

4246  1  221 

197 

4466 

4687 

4907 

5.27 

5347 

5567 

5787 

6007 

6226 

6446  i  220 

198 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

2.9 

199 

8853 

9071 

9289 

9007 

9725 

9943 

•161 

•378 

•595 

•8i3 

2I8| 

200 

3oio3o 

1247 

1464 

1681 

1898 

2114 

233i 

2047 

2764 

2980 

217 

201 

3196 

3412 

3628 

3844 

4039 

4275 

449  J 

4706 

4921 

5i36 

216 

202 

535, 

5566 

5781 

59q6 

62.1 

6425 

6639 

6854 

7068 

7282   2l5| 

203 

749^' 

7710 

7924 

8,37 

835i 

8564 

8778 

8991 

9204 

9417  !  2i3| 

204 

9630 

9^43 

••56 

•268 

•481 

•693 

•906 

1118 

i33o 

i542  i  212I 

2o5  311754 

1966 

2177 

23V9 

2600 

2812 

3o23 

3234 

3445 

3656 

211 

206 

3867 

4078 

4289 

4499 

47i'o 

4920 

5i3o 

5340 

555i 

5760 
7854 

2.0! 

Tol 

5970 

6180 

6390 

6.-)99 

6809 
8898 

7018 

7227 

7436 

7646 

200  1 

8o63 

^272 

84?Si 

Sb^g 

9106 

9314 

9522 

9730 

9938   2081' 

209  i32oi46 

o3j4 

o562 

0769 

0977 

1 184 

1391 

098 

i8o5 

2012 

207 

210  |3222lg 

2426 

2633 

2839 

3046 

3252 

3458 

3665 

3871 

4077 

206 

211 

4282 

4488 

4694  ;  4899 

5io5 

53io 

55 1 6 

5721 

5926  I  6i3i 

2o5 

212 

6336 

6541 

6745   6950 

7.55 

7359 

7563 

77(>7 

7972   8176 

204 

2l3 

8380 

85o3   8787  i  899. 

9 '94 

9398 

9601 

9S05 

•••8   •211 

203 

214  330414 

0617 

0819   1022 

1225 

1427 

i63o 

lt*32 

2o34  2236 

202 

2i5   2438  2640 

2842  1  3o44 

3246  -3447 
5257  I  5458 

3649 

385o  1  4o5i   4253 

202 

216   4454  4655 

4856  j  5o57 

5658 

5859  :  6059  1  6260 

201 

217   6460  6660 

6S60   7060 

7260  ,  745q 

7659 

7858  ;  8o58  ;  8257 

200 

218   8456  8656 

8-S55  1  9054  9253  ]  945 1  ! 

9630  ' 

0849  ;  ••47  "246 

199 

219  340444  0642  1  0841   1039  1  1237  1  1435  1 

i632  J  i83o  1  2028  '   2225  1 

198 

N.  1  0 

I   1   2   j   3   j  4   1   5   1  6   1   7   1  8   1   9   j  1). 

15 


A   TABLE    OF   LOGARITHMS   FEOM   1    TO   10,000. 


IS\  j   o     I 

.   i  3 

4 

5 

' 

7 

8 

9 

D. 

220  342423  2620 

2817   3oi4 

3212 

3409 

36o6 

38o2 

3999 

4196 

197 

221    4392  4589 

4785  4931 

5178 

5374 

5570 

5766 

5962 

6157 

196 

222   6353  6549 

6744  6939 

7135 

7330 

732D 

7720 

7913 

8110 

195 

223  !  83o5  85oo  1 

8694  8889 

9083 

9278 

9472 

9666 

9^60 

••34 

194 

224  350248  0442 

o636  1  0820 

1023 

1216 

1410 

i6o3 

1796 

1989 

193 

.     225 

2i83  2375  '  2568  !  2761 

2954 

3i47 

3339 

3532 

3724 

3916 

193 

226 

4108  43oi 

4493  1  4685 

4876 

5o68 

5260 

5452 

5643 

5834 

192; 

227 

6026  6217 

6408  1  6099 

6790 

6981 

7172 

7363 

7554 

7744 

19! 

228 

7935  8125 

83 16  1  85o6 

8696 

8886 

9076 

9266 

9456 

9646 

190. 

i  229 

9835  ••25 

•2l5 

•404 

•593 

•7S3 

•972 

1161 

i35o 

i539 

189 

23o 

361728  1917 

2io5 

2294 

2482 

2671 

2859 

3o48 

3236 

3424 

188 

23l 

3612  38oo 

3988 

4176 

4363 

455i 

4739 

4926 

5ii3 

53oi 

188 

232 

5488  5675 

5862  1  6049 

6236 

6423 

6610 

6796 

6983 

7169 

187 

233 

7356  7^42 

7729 

791 D 

8101 

8287 

8473 

8639 
•5i3 

8845 

9o3o 

186 

234 

9216  9401 

9587 

9772 

9958 

•143 

•328 

•698 

•883 

i85 

235 

371068  1253 

1437 

1622 

1806 

1991 

2175 

236o 

2544 

2728 

184 

236 

2912  3096 

3280 

3464 

3647 

3s3i 

4013 

4.98 

4382 

4565 

184 

237 

4748  4932   5ii5 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

1 83 

238 

6577  6759  6942 

7124 

7306 

7488 

7670 

7852 

8o34 

8216 

182 

239 

8398^  858o  ;  8761 

8943 

9124 

9306 

9487 

9668 

9349 

••3o 

181 

240 

38021 1 '  0392  !  0573 

0754 

0934 

ni5 

1296 

1476 

1 656 

1837 

181 

241 

2017'  2197  !  2377   2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

242 

38 1 5  3995  1  4174  1  4353 

4533 

4712 

4891 

5070 

5249 

5428 

179 

243 

56o6  5785  |  5964 

6142 

6321 

6499 

6677 

6856 

7034 

7212 

178 

1  244 

7390'  7363  7746 

7923 

8101 

8279 

8436 

8634 

8811 

8989 

178 

1  245 

9166;  9343   9^20 

9698 

9875 

••5i 

•228 

•4o5 

•582 

•739 

177 

246 

390935:  1 1 12  ;  1288  1  1464 

i64i 

1817 

1993 

2169 

2345 

2321 

176 

247 

2697:  2^73  1  3043  3224 
4452'  4627  4802  4977 

3400  t  3575 

3731 

3926 

4101 

4277 

176 

248 

5i52 

5326 

55oi 

5676 

585o 

6025 

175 

249 

6199'  6374  6548  1  6722 

6896 

7071 

7245 

7419 

7592 

7766 

174 

25o 

397940!  8114  8287  1  8461 

8634 

8808 

8981 

9154 

9328 

95oi 

173 

25l 

9674  9?47  1  ••20  1  ^192 

•365 

•538 

•711 

•883 

io56 

1228 

173 

252 

401401:  1573  !  1745  1  1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

253 

3 121  3292   3464   3635 

3807 

3978 

4149 

4320 

4492 

4663 

171 

254 

4834  5oo5   6176   5346 

5517 

5688 

3836 

6029 

6199 

6370 

171 

255 

6540.  6710  1  6881  1  7o5i 
8240  841 0  j  8579  I  8749 

7221 

7391 

7561 

7731 

7901 

8070 

170 

256 

8918 

9087 

9237 

9426 

9595 

9^64 

169 

257 

9933:  •102  i  •271  1  •440 

•600 
2293 

•777 

•946 

1114 

1283 

I45i 

169 

258 

41 1020'  1788  j  1956  ]  2124 

2461 

2629 

2796 

2964 

3i32 

168 

259 

33oo!  3467  3635 

38o3 

3970 

4i37 

43o3 

4472 

4639 

4S06 

167 

260 

414973!  5i40   5307 

5474 

5641 

58o3 

5974 

6141 

63o8 

6474 

167 

261 

6641  6807   6973 

7i39 

73o6 

7472 

7638 

7804 

7970 

8i35 

166 

262 

83oi!  8467  '  8633  1  8793 
9906  •ni  1  •286  ;  •45i 

8964 

9129 

9295 

9460 

9625 

9791 

1 65 

263 

•616 

•781 

•945 

IIIO 

1275 

1439 

i65 

264 

421604!  1788   1933 

2097 
3737 

2261 

2426 

2390 

2754 

2918 
4555 

3082 

164 

265 

32461  3410  3574 

3901 

4o65 

4228 

4392 

4718 

164 

266 

4SS2'  5o45  '  5208 

5371 

5534 

0697 

586o 

6023 

6186 

6349 

1 63 

267 

65ii!  6674  1  6836 

6999 

7161 

7324 

7486 

7648 

781 1 

7973 

162 

268 

8135:  8297  1  8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

269 

9752,  9914  •^s 

•236 

•398 

•559 

•720 

•881 

1042 

1203 

161 

270 

43i364'  i525  :  i685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

271 

2969'  3i3o  3290 

3450 

36io 

3170 

3930 
5526 

4090 

4249 

4409 

160 

272 

4569  4729  4888  j  5048 

5207 

5367 

5685 

5844 

6004 

\? 

273 

6i63  6322  1  6481   6640 

6793 

6957 

7116 

8859 

7433 

7592 

I58 

274 

775i!  7909  '   8067  !  8226 

8384 

8542 

8701 

9017 

9175 

275  1  9333;  9491  1  9648  i  9806 

??^i 

•122 

•279 

•437 

•594 

•752 

276  I440909'  1066 

1224  1  i33i 

1695 

i852 

2009 

2166 

2323 

1 57 

277  !  24S0  2637 

2793   2950 
43D7  45i3 

3 106 

3263 

3419 

3576 

3732 

3889 

'i'^ 

278   4045  4201 

4669 

4825 

4981 

5x37 

5293 

5449 

1 56 

279  ;  56o4|  5760 

5915  6071 

6226 

6382 

6537 

6692 

6848 

7003 

i55 

1  N. 

1 

0 

I 

2   1  3 

4 

5 

6 

7 

B 

9 

D. 

A  TABLE   OF   LOGARITHMS   FROM   1    TO    10,000. 


N. 

0 

I 

1   ^ 

3 

4 

1  ' 

6   1   7 

8     9 

,  D. 

280 

447158 

73i3 

7468 

7623  7778 

7933 

8088 

8242 

8397 

8552 

;  l55 

281 

8706 

8861 

90i5 

9170  9324 

9478 

9633 

9787 

9941 

,  ^^95 

:  i54 

282 

430249 

o4o3 

0557 

0711 

0865 

1018 

1172 

i326 

1479 

:  1633 

i54 

283 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3oi2 

j  3.65 

1  i53 

284 

33i8 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

1  4692 

!  i53 

285 

4845 

4997 

5i5o 

53o2 

5454 

56o6 

5758 

5910 

6062 

■  6214 

1  l52 

286 

6366 

65i8 

6670 

6821 

6973 

7125 
8638 

7276 

7428 

7^79 

1  773. 

!  132 

287 

7882 

8o33 

81 84 

8336 

8487 

8789 

8940 

9091 

1  9242 

1  i5i 

288 

9392 

9543 

9694 

9845 

9995 

•146 

•296 

•447 

•597 

•748 

1  i5i 

289 

460898 

1048 

1 198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

i5o 

290 

462398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 
5o85 

3744 

i5o 

291 

3893 

4042 

4191 
568o 

4340 

4490 

4639 

4788 

4936 

5234 

149 

292 

5383 

5532 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

293 

6S68 

7016 

7164 

7312 

7460 

7608 

7756  1  7904 

8o52 

8200 

148 

294 

8347 

8495 

8643 

8790 

8938 

9085 

9233  1  9380 

9527 

9675 

148 

295 

9822 

9969 

•116 

•263 

•410 

•557 

•704 

•85 1 

•998 

1145 

147 

2q6 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

23i8 

2464 

2610 

146 

297 

2756 

2903 

3o49 

3195 

3341 

3487 

3633 

3779 

It 

407. 

146 

29S 

4216!  4362 

45o8 

4653 

4799 

4944 

5090 

5235 

5526 

.46 

299 

5671 

58i6 

5962 

6107 

6202 

6397 

6542 

6687 

6832 

6976 

145 

3oo 

477121 

7266 

741 1 

8855 

s 

7700 

7844 

7989 

8i33 

8278 

8422 

145 

3oi 

8566 

871 1 

9143 

9287 

943 1 

9575 

9719 

9863 

144 

302 

480007 

oi5i 

0294 

o582 

0725 

0869 

1012 

ji56 

1299 

144 

3o3 

1443:  1 586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

3  04 

2874  3oi6 

3i59 

3302 

3445 

3587 

3730 

3872 

401 5 

4.57 

143 

3o5 

43oo  4442 

4585 

4727 

4869 

5oii 

5i53 

5295 

5437 

5579 

142 

3o6 

572 1 1  5863 

6oo5 

6.47 

6289 

643o 

6572 

6714 

6S55 

6997 

142 

3o7 

7i3Si  7280 

7421 

7563 

7704 

7845 

7986 

8.27 

8269 

84.0 

141 

3  08 

855i 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

.41 

3o9 

9958 

••99 

•239 

•38o 

•520 

•661 

•801 

•941 

1081 

1222 

149 

3io 

491362 

l502 

1642 

1782 

1922 

2062 

2201 

2341 

2481 

2021 

140 

3ii 

2760 

2900 

3  040 

3179 

3319 

3458 

3597 
4989 

3737 

3876 

40.5 

i3o 

3l2 

41 55 

4294 

4433 

4072 

471 1 

485o 

5128 

5267 

5406 

.39 

3i3 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

65i5 

6653 

6791 

.39 

3 14 

t^° 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8o35 

8173 

138 

3i5 

8448 

8586 

8724 

8862 

8999 
•374 

9.37 

9275 

9412 

9550 

i38 

3i6 

9687 

9824 

9962 

••99 

•236 

•5n 

•648 

•785 

•922 

.37 

3i7 

5oio59 

1 196 

i333 

1470 

1607 

1744 

1880 

2017 

2i54 

2291 

137 

3i8 

2427J  2564 

2700 

2837 

2973 

3109 

3246 

3382 

33i8 

36d5 

i36 

3.9 

3791  3927 

4o63 

4199 

4335 

4471 

4607 

4743 

4878 

5oi4 

1 36 

320 

5o5i5o 

5286 

5421 

5557 

5693 

5828 

5964 
73i6 

6099 

6234 

6370 

i36 

321 

65o5 

6640 

6776 

69U 

7046 

7181 

745i 

7586 

7721 

i35 

322 

7856 

7991 

8126 

8260  8395 

853o 

8664 

8799 

8934 

9068 

i35 

0  23 

9203 1  9337 

9471 

9606  i  9740 

9874 

•••9 

•143 

•277 

•4.1 

i34 

324 

5io545  0679 

o8i3 

09^7  1  loSi 

I2l5 

1 349 

1482 

1616 

1750 

1 34 

325 

1 883 

2017 

2l5l 

2284 

2418 

255i 

2684 

2818 

2951 

3o84 

i33 

326 

3218 

335i 

3484 

3617 

3750 

3883 

4016 

4149 

4282 

4414 

i33 

327 

454S 

4681 

48i3 

4946 

5079 

5211 

5344 

5476 

5609 

5741 

i33 

328 

5874i 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

l32 

329 

7196;  7328 

7460 

7592 

7724 

7855 

7987 

8119 

825i 

8382 

l32 

33o 

5i85i4|  8646 

8777  1 

8909  1  9040 

9171 

93o3 

9434 

9566 

9697 

i3i 

33 1 

9828;  9959 

••90  1 

•221  i  ^353 

•484 

•6i5 

•745 

•876 

1007 

i3i 

332 

52ii38i  1269 

1400  I 

i53o 

1661  1 

1702 

1922 

2o53 

2.83 

23i4 

i3i 

333 

24441  2575   2705  ! 

2835 

2966  1  3096 

3226 

3356 

3486 

36.6 

i3o 

334 

3746  3876  4006  : 

4i36 

4266  4396 

4526 

4656 

4785 

49' 5  i 

i3o 

335 

5o45;  5 1 74 

53o4 

5434 

5563   5693   5822  I  5q5i  | 

60S  I 

6210 

129 

336 

6339'  6469 

^^2^  i 

6727 

6856 

6985   7114  t  7243 

7372 

75oi 

129 

337 

763o  7759 

7888 

8016 

8145 

8274  I  8402  '   853 1 

8660 

8788 

\lt 

338 

8917,  9045  9174  '  9302 

943  0 

9559  1  9687  ;  9^i5 

9943 

•072 

339 

530200  0328  0456  o584 

0712 

0840  0968   1096 

1223 

i35i 

128 

N. 

0  1   . 

2 

3 

4  1   5   1   6  1 

7   1 

8 

9 

D. 

A   TABLE    OF    LOGARITHMS   FROM    1    TO   10,000. 


N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

340 

53I4T9  1607 

1734 

1S62 

199  J 

21 17   2245 

2872 

23oo   2627 

I  28 

341 

27j^  2io2 

8009 

3i36 

3264 

3391  1  33i8 

3645 

3772 ;  3.899 

127 

342 

402^  41 53 

42  >o 

4407 

4534 

4661  i  4787 

4914 

5o4i  1  5.07 

127 

343 

529*  5421 

5047 

5074 

5800 

5927  6o33 

6180 

63o6  ,  6432 

12U 

344 

6jj8  6j35 

6m  I 

6937 

7063 

7189 

7810 

7441 

7567  i  7693 

126 

34J 

7S19  7Q45 

8071 

8197 

8322 

8448 

8074 

8O99 

b625    >    8901 

120 

3^6 

9070  9202 

9327 

9432 

9578 

97o3 

9S29 

9954 

••79   ^204 

125 

347 

540 J 29  0430 

OJOO 

0700 

o83o 

0.^00 

J  0^0 

I205 

.3 Jo   1434 

.23 

3-,6 

IJ79  1704 

1>>29 

1953 

2078 

22o3 

2J27 

2452 

2376    270. 

.25 

349 

2020  2900 

3074 

3199 

3323 

3447 

357i 

3096 

3820  1  3944 

.24 

35o 

544068  4192 

43 16 

4440 

4564 

4688  1  4812 

4936 

5o6o  1  5i83 

124 

3Ji 

5 Jo  7  5481 

5555 

5678 

5802 

5920  :  6049 

6172 

6296  !  64.9 

124 

352 

65^3  bboO 

6769 

69,3 

7086 

7159  ;  7202 

74o5 

7529 

7652 

123 

3J3 

7770  7-98 

8021 

8144 

8267 

8889  ;  8012 

8635 

8758 

8881 

.23 

354 

gooi    9126 

9249 

9371 

9494 

9616  ,  9789 

9861 

9984 

•106 

.23 

330 

55022S  oJ5i 

0.73 

0595 

0717 

0840  ;  0902 

1084 

.206 

1328 

.22 

350 

1400  1072 

1694 

I8i6 

1988 

2060  \   21:31 

23o3 

2425 

2547 

122 

357 

200j  2-90 

291 1 

3o33 

3.55 

3276 

3398 

33.9 

3640 

3762 

.2. 

35tf 

3:i>3  4004 

4126 

4247 

4368 

44.89 

4010 

4731 

4852 

4973 

12. 

359 

5094  52 1 5 

5336 

5457 

5578 

5099 

30.20 

0940 

6061 

6182 

12. 

36o 

5563o3  6423 

6544 

6664 

67S5 

6905 

7026 

7146 

7267 

7387 

120 

361 

7507  7527 

7748 

7068 

7988 

8108 

&228 

8349 

84^>9 

85^9 

.20 

362 

8709  8  29 

8943 

9068 

9188 

9808 

9428 

9548 

96O7 

9787 

.20 

363 

99 J7  ••26 

•146 

•2b5 

•335 

•5o4 

•624 

•743 

i863 

•982 

1.9 

364 

56iior  1221 

1 340 

1459 

1078 

1698 

KM  7 

.986 

2o55 

2174 

1.9 

360 

2298  2412 

233 1 

263o 

2769 

2S87 

3oo6 

3.25 

3244 

3362 

.19 

306 

3^oi;  35oo 

3718 

3837 

3955 

4074 

4192 

43.1 

4429 

4548 

119 

367 

466d,  4704 

49^3 

502I 

5.39 

5257 

D8-6 

5494 

56.2 

5780 

u>> 

36d 

5b48|  5966 

60..4 

6202 

6320 

6437 

O3o5 

6673 

6191 

6909 

118 

369 

70261  7144 

7262 

7379 

7497 

7614 

7782 

7849 

7967 

8084 

u8 

370 

56S202;  83 1 9 

8436 

8554 

8671 

8788 

8905 

9023 

9.40 

9257 

H7 

371 

9374'  9491 

9608 

9725 

9842 

9959 

••76 

•.93 

•309 

•426 

'n 

i^^ 

570548  0060 
170^9'  I~>25 

0776 

0J93 

1010 

1126 

1243 

1339 

1476 

.5q2 

117 

373 

1942 

2o58 

2174 

229I 

2407 
3368 

2523 

2689 

2755 

116 

374 

2.->72  2g83 

3 104 

3220 

3336 

3452 

3684 

3800 

39,5 

116 

1  373 

40  J 1 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

0072 

116 

:  3-0 

5id8 

53o3 

5419 

5534 

5650 

5765 

D8bo 

5996 

6. II 

6226 

1.5 

377 

634! 

6407 

6572 

6687 

6802 

69,7 

7082 

7147 

7262 

7377 

1.5 

37^ 

74921  7607 
8689  8704 

7722 

7836 
8983 

7951 

8006. 

8181 

8290 

841 0 

8525 

1.5 

379 

8068 

9097 

92.2 

9826 

9441 

9555 

9669 

114 

3^0 

5797«4'  9s^98 

••12 

•126 

•241 

•355 

•469 

•583 

:^i 

•811 

114 

38! 

58092O:  1089 

n53 

1267 

i38i 

1495 

1608 

1722 

1950 

114 

3S2 

2oj3-  2177 

2291 

2404 

25i8 

263 1 

2T45 

2858 

2972 

3o85 

114 

383 

3199'  33i2 

3.26 

3539- 

3652 

3765 

3079 

3992 

4io5 

4218 

1.3 

384 

4381 

4444 

4357 

4670 

4783 

4896 

0009 

5,22 

5235 

5348 

1.3 

380 

5461 

5374 

5o86 

5799 

5912 

6024 

6.37 

6250 

6362 

6475 

ii3 

356 

65:>7 

6700 

bSi2 

6925 

7087 

7U9 

72&2 

7374 

7486 

7599 

112 

iV. 

77" 

7S23 

7935 

8047 

8160 

8272 

8384 

8496 

8608  8720 

112 

388 

8o32  ,SgU 

9j36 

9167 

9279 

9391 

93o3 

g'MO 

9726  '  9'i38 

112 

339 

9900^  ••bi 

•173 

•284 

•396 

•5o7 

•619 

•780 

•842 

•953 

112 

390 

591065  1 176 

1287 

•399 

i5io 

i(  Vi 

1732 

1843 

1955 

2066 

III 

391 

2177I  22^8 

2399 

2010 

262 1 

2:32 

2843 

2904 

3o64 

3i75 

11. 

392 

3286;  3397 

33o8 

36i8 

3729 

3840 

3q5o 

4061 

4171 

4282 

.11 

393 

4893  45oJ 

4614 

4724 

4834 

4945 

5o35 

5.65 

5276  5386 

no 

394 

5496  56o6 

5717 

5827 

5937 

6047 

6,57 

6267 

6377 

6487 

no 

390 

6597  6707 

68.7 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

no 

396 

7690,  73o5 
8791!  8900 

79>4 

8024 

8i34 

8243 

8353 

8462 

85-2 

8681 

no 

397 

9009 

9119 

9228 

9337 

9446 

9356 

9665 

9774 

109 

398 

9883  9r.r.2 

•lOI 

•210 

•3.9 

•428 

•537 

•646 

•755 

•864 

109 

399  603973   10^2 

IIQI 

1299 

1408 

i5i7 

1620 

1734 

1843 

195. 

109 

j  N.  i  0 

I 

2 

3 

4 

5   i  6 

7 

8 

9 

D. 

A  TABLE   OF   LOGARITHiES   FRO:S\.   1    TO    10,000. 


N. 
400 

0 

1 

I       2 

3   j  4   !  5 

6:7.819 

D. 

|)02060 

2169   2277 

2386 

2494 

1  2603 

2711 

1  2819   2928  I  3o36 

108 

401 

;  314/ 

3253   336i 

3469 

3377 

1  3686 

3794 

3902  '  4010  1  4118 

108 

402 

{  4226 

4334  4442 

455o 

4658 

1  4766 

4874 

1  4982  :  5089  1  D197 

108 

4o3 

53o5 

54«3 

D52I 

5628 

D736 

i  5844 

5951 

,  0039  6166  1  6274 

,  'o^i 

404 

63oi 

6489  6J96 

6704 

6811 

I  6919 

7026 

:  7133  7241  :  7348 

1  107! 

4o5 

7455 

7062   7609 

7777 

7884 
8954 

:  7991 

8098 

!  8203 

;  8312 

!  84«9 

i  107 

406 

8026 

8633   8740 

8847 

j  9061 

9167 

1  9274 

1  9381 

94»8 

1   '  1 
j  107' 

407 

9594 

9701   9808 

9914 

••21 

!  •128 

•234 

•341 

•447 

•554 

i  '07 

408 

01 0660 

07^67   0873 

0979 

10S6 

1 1192 

1298 

'  i4o5 

1311 

16,7 

1  106 

409 

1723 

1829  1  1936 

2042 

2148 

2254 

236o 

;  2466 

12572 

^678 

i  'o6| 

410 

612784 

2890   2996 

3l02 

3207 

33 1 3 

3419 

i  3325 

3630 

3736 

1  106, 

411 

3842 

3947 

4o53 

4 1 59 

4f64 

4370 

4475 

1  4581 

46% 

4792 

;  io6; 

412 

4897 

5oo3 

5io8 

3213 

5319 

3424 

5529 

1  5634 

5740 

5845 

;  io5' 

4i3 

5q5o 

6o55 

1  6160 

6265 

6370 

6476 

658 1 

;  6686 

6790 

6895 

i  io5i 

414 

7000 
8048 

7.o5 

.  7210 

73i5 

7420 

7525 

7629 

7734 

78J9 

7943 

'  100' 

4i5 

8i53 

,  ^"7 

8362 

8466 

!  »37I 

8676 

:  8780 

1  8884 

89H9 

103 

416 

9093 

9.98 

9302 

9406 

95.1 

;  9615 

9719 

9^24 
'  0S64 

I  9928 

••32 

i  104 

417 

620 I J6 

0240 

i  0344 

0448 

0332 

i  o656 

0760 

1  0968 

1072 

!  104; 

418 

1 1 76 

J  280 

i  1384 

1488 

1592 

j  1695 

1799 

,  1903 

i  2007 

2110 

i  104; 

419 

2214 

23i8 

j  242. 

2325 

2628 

;  2732 

2835 

2939 

3o42 

3146 

i  104 

1  420 

623249 

3353 

3456 

3559 

3663 

!  3766 

3S69 

3973 

4076 

4179 

103  i 

i  421 

42b2 

4385 

!  4488 

4591 

4693 

i  4793 

4901 

5oo4 

5107 

5210 

io3, 

422 

53i2 

54i5 

55i8 

3621 

5724 

3827 

5929 

6o32 

6i35 

6238 

103! 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7o58 

7161 

7263 

1  io3; 

424 

7366 

7468 

7571 

7673 

777^ 

7«78 

7980 

8082 

8i85 

8:87 

i  102 

1  423 

8389 

8491 

8593 

8695 

«797 

8900 

9002 

9104 

9206 

930S 

102: 

426 

9410 

95i2 

9613 

9713 

9817 

9919 

••21 

•123 

•224 

•326 

102 

^'Z 

630428 

o53o 

o63i 

0733 

oS33 

0936 

io38 

.139 

1241 

1 342 

102 

428 

1444 

1 545 

1647 

1748 

1849 

1951 

2052 

2133 

2235 

2356 

lOI 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3i65 

3266 

3367 

101  i 

43o 

633468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

( 
100 

43 1 

4477 

4578 

4679 

4779 

4880 

4981 

5o8i 

5182 

5283 

5383 

100 

432 

5484 

5584 

3685 

5785 

5886 

5986 

6087 

6187 

6287 

6388 

100  j 

433 

6488 

6588 

6688 

67S9 

6889 

6989 

7089 

7.89 

7290 

83?9 

100! 

434 

7490 

7590 
8589 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

99 

435 

8489 

86b9 

8789 

88a8   89.^8 

9088 

9188 

9287 

9387 

99 

436 

9^^6 

9586 

9680 

9783 

9885   99S4 

••84 

•i83 

•2SJ 

•382 

99 

^?Z 

640481 

o58i 

0680 

0779 

0879   0978 

1077 

1177 

1276 

1375 

99 

1  438 

1474 

.573 

1672 

1771 

1871 

1970 

2069 
3o58 

2i68 

2267 

2366 

99 

439 

2  465 

2563 

2662 

2761 

2860 

2939 

3i36 

3255 

3334 

99 

440 

643453 

355i 

365o 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

98 

441 

4439! 

4537 

4636 

4734 

4832 

493. 

5029 

5x27 

5226 

5324 

98' 

442 

5422 

5521 

36l9 

5717 

38i5 

5913 

601 1 

6110 

6208 

63o6 

98 

1  443 

6404  65o2 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

!  444 

7383!  7481 

2^19 

7676 

7774 

7872 

8943 

8067 

8i65 

8262 

98: 

i  445 

836o  8458 

8555 

8653 

8750 

8848 

9043 

9140 

9237 

97 

;  446 

9335  9432 

953o 

9627 

9724 

9821 

0890 

••16 

•ii3 

•210 

97 

447 

65o3o8  o4o5 

0502 

0599 

0696 

0793  , 

0987 

1084 

1181 

97 

448 

.278  1375 

1472 

1 569 

I6b6 

1762 

1809 

1936 

2o53 

2i5o 

97 

449 

2246 

2343 

2440 

2536 

2633 

2730  1 

2826  : 

2923 

3019 

3ii6  j 

97 

450 

653213' 

3300 
4273 

34o5 

35o2 

3598 

3695 

3791  1 
4734  ! 

3888 

3984 

4080  i 

96 

431 

4«77' 

4369 

4465 

4562 

4658 

485o 

4946 

5o42  1 

96 

452 

5i38;  5235  | 

5331 

5427 

5523 

3619  1 

5715  :  58io  ! 

5906 

6002  i 

96 

453 

609S 

6194  1 

71^2  1 

6290 

6386 

6482 

^?77 

6673  6769  j 

6864  , 

6960  1 

96 

454 

7036 

7247 

7343 

7438 

7334 

7629  7725  I 

7820  . 

7916  1 

96, 

455 

Son.  8107 

8202   8298 
9155  1  925o 

8393 

8488 

8584  )  8679  8774  ! 

8870  1 

95 

456 

8965:  9060 

9346 

9441 

9536  9631   9726  j 

9821 

95; 

457 

9916  ••!! 

•106   •201 

•296 

•391 

•4St»   •58i   •676  ■  •771  i 

95 

458  66o665  0960 

io55 

ii5o 

1245 

1339 

14U   i529  1  1623   1718  ; 

95: 

459   i8i3,  1907 

2002 

2096 

2191 

2286 

238o  1  2473 

2569  i 

2663  j 

95 

N. 

0 

» 

2 

3 

4 

5 

6     7 

8  ! 

N 

A  TABLE  OF  LOGARITHMS  FROM:  1  TO  10,000. 


» 

460 

012     3 

4   1   5 

6 

' 

8 

9 

D. 

662-;3  2S52  j  2947   3o4i 

3i35  323o 

3324  1  3418 

35i2 

3607 

94 

461 

3701  3795   3od9  3933 

4078 

4172 

4266 

436o 

4434 

4548 

94 

462 

4642  4736   453o   4924 

5oi8 

5lI2 

D206 

5299 

5393 

5487 

94 

463 

55Si  5675   5769  5862 

3936 

6o5o 

6143 

6237 

633 1 

6424 

94 

464 

65 1 8  6612  1  6700  6799 

6:592 

6986 

JSi? 

7.73 

7266 

7360 

94 

465 

7453  7546  1  7640  7:33 

7826 

7920 

S106 

8.99 

8293 

93 

466 

8386  8479  '  ^^1^      ^^^ 

8739 

8852 

8945 

9038 

9i3i 

9224 

93 

n 

9317  9410  1  9^0^   9^9^ 

9689 

9782 

9873 

9967 

••60 

•i53 

93 

670246 

0339  '  043 1   OJ24 

Otil-] 

i543 

0710 

0802 

0895 

0988 

loSo 

9^ 

469 

..73 

1263   i35:i   I40I 

1636 

1728 

1821 

1913 

2005 

93 

470 

67209S 

2190 

2233   2375 

2467 

256o 

2652 

2744 

2836 

2929 

92 

47 « 

302I 

3ii3 

320J  3297 

3390 

34i<2 

3574 

3666 

3758 

385o 

92 

472 

3942 

4j34 

4120  ;  4218 

43io 

4402 

4494 

4586 

4677 

4769 

92 

473 

4H6. 

4953 

3045  1  5i37 

3228 

5320 

5412 

55o3 

5395 

5087 

92 

'  474 

5778 

5':>70 

5962  ;  6o53 

6145 

6236 

6328 

6419 

65 11 

6602 

92 

;  475 

6694 

67«5 

6576 

6968 

7039 

-7.5. 

7242 

7333 

7424 

7516 

9» 

476 

7607 

7t>98 

7759 

7881 

79^2 

8o63 

81 54 

8245 

8336 

8427 

91 

477 

85iS 

8609 

8700 

879' 

8582 

8973 

9064 

9155 

9246 

9337 

9» 

47' 

942-1  9319 

9610 

9700 

979' 

9882 

9973 

••63 

•i54 

•243 

91 

;  479 

68o336  0426 

03I7 

0607 

0698 

0789 

0879 

0970 

1060 

ii5i 

91 

1 
1  4S0 

681 241 

i332 

1422  j  i5i3 

i6o3 

1693 

1784 

1874 

1964 

2o55 

90 

4M 

2145 

2235 

2326   2416 

25o6 

2596 

2686 

2777 

25.67 

2937 

90 

482 

3o47i  3 1 37 

3227  :  3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

3947;  4037 

4127  4217 

4307 

4396 

4486 

4576 

4666 

4756 

90 

484 

4845:  4935 

5o23  ,  5ii4 

5204 

5294 

5383 

5473 

5563 

5652 

90 

;  485 

5742!  5d3i 

5921  :  6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

486 

6636|  6726 

681 5  ■  6904 

6994 

7o83 

8^64 

7261 

735i 

7440 

89 

,  487 

75291  7618 
842oi  a5o9 

7707  !  7796 

7836 

7975 

8i53 

8242 

8331 

89 

:  488 

6093  ,  bt>b7 

8776 

8865 

8933 

9042 

913. 

9220 

89 

:  489 

9309!  9398 

9486  ,  9575 

9664 

9733 

9841 

9930 

••19 

•107 

89 

1  490 

690196 

0285 

0373  j  0462 

o55o 

o639 

0728 

0816 

0905 

0993 

89 

j  491 

lOdi 

1 1 70 

1238  j  1347 

1435 

1 524 

1612 

1700 

'7-9 

1877 

88 

1  492 

1965 

2oj3 

2142 

223o 

23i8 

2406 

2494 

2583 

2671 

2759 

88 

1  493 

2847 

2o35 
38i5 

3o23 

3iii 

3199 

3287 

3375 

3463 

3551 

3639 

88 

494 

3727 

3903 

3991 

407s 

4166 

4234 

4342 

443o 

4517 

88 

495 

46o5 

4693 

4781 

48t»8 

4956 

5o44 

5i3i 

5219 

53o7 

5394 

88 

496 

5482 

5509 

5037 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

497 

6356 

6444. 

633 1 

66i3 

6706 

6-93 

6SS0 

6968 

7o55 

7142 

87 

498 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

499 

8101 

bitid 

8275 

83b2 

8449 

8535 

8622 

8709 

8796 

8883 

87 

5oo 

698970 

9057 

9144 

923 1 

93.7. 

9404 

9491 

9378 

9664 

975 1 

87 

DO  I 

9"^38 

9924 

••11 

.•98 

•184 

•271 

•358 

•444 

•53 1 

•617 

87 

502 

700704!  0793 

0877  0963 

io5o 

ii36 

1222 

.309 

1395 

1482 

86 

5o3 

i568i  i6d4 

1741  !  1827 

1913 

1999 

2086 

2172 

2208 

2344 

86 

5o4 

2431 

2317 

2603  ,  2689 

2775 

2861 

2947 

3o33 

3ii9 

32o5 

86 

5o5 

3291 

3377 

3463  1  3549 

3635 

3721 

3807 

3893 

39-9 

4o65 

86 

5o6 

4i5i 

4236 

4322  j  4408 

4494 

4579 

4665 

475i 

4837 

4922 

86 

507 

5oo8:  5094 

3179  i  ^265 

5350 

5436 

5522 

56o7 

5093 

5778 

86 

5o8 

5864 ;  3949 

6o35  1  6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 

6718!  6So3 

6888  6974 

7059 

7>44 

7229 

73i5 

7400 

7485 

85 

5io 

707570I  7655 

7740  7826 
8591   8676 

791 1 

7996 

8081 

8166 

825i 

8336 

85 

5ii 

84211  85o6 

8761 

8846 

8931 

9015 

9100 

9185 

85 

5l2 

9270  9355 

9440  1  9524 

9609 

9694 

9779 

9863 

9948 

••33 

85 

5i3 

710117  0202 

0287  ;  0371 

0456 

0540 

0625 

0710 

0794 
1 639 

0879 

85 

5i4 

0963,  1048 

ii32   1217 

i3oi 

1 385 

1470 

1554 

1723 

84 

5:5 

1807  1892 

1976  i  2060 

2144 

2229 

23.3 

fil 

2481 

2566 

84 

5i6 

265o  2734 

2818  1  2902 

29% 

3070 

3.54 

3323 

3407 

84 

5,7 

3491  3575 

3659  j  3742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

5i8 

433o  4414 

4497   458 1 

4665 

474Q 

4833 

4916 

5ooo 

5o34 

84 

519 

5167,  525i 

5335  1  5418 

55o2  1  5586 

5669 

5753 

5836 

5920 

84 

N. 

i  «  i  ' 

a 

3 

.  1  5 

' 

7 

« 

y 

D.i 

A  TABLE   OF   LOGARITHMS   FROM   1    TO    10,000. 


N. 

0 

I 

2 

3    1    4 

5    6 

7     8  ;  9 

D. 

520 

716003 

6087 

6170 

6254 

6337 

6421  i  65o4 

6588  6671  1  6754 

83 

521 

6838 

6921 

7004 

7088 

7171 

7254  1  7338 

7421   7504  7387 

83 

522 

7671 

7754 

7837 

7920 

8oo3 

8086  1  8169 

8253   8336 

1  8419 

83 

523 

85o2 

8585  j  8608 

8751 

8834 

8917  1  9000 

9083   9165 

9248 

83 

524 

9331 

9414  :  9497 

9380 

9663 

9745  1  9828 

99 1 1   9994 

••77 

83 

525 

720159 

0242  ,  o325 

0407 

0490 

0573 

0633 

0738  0821 

0903 

83 

526 

0980 

1068  ,  ii5i 

1233 

i3i6 

1398 

1481 

1 563 

1646 

1728 

82 

527 

1811 

1893 

1975 

2o58 

2140 

2222 

23o5 

2387 

2469 

2552 

82 

52y 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

82 

529 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4o3o 

4112 

4194 

82 

53o 

724276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

4931 

5oi3 

82 

53 1 

5095 

5176 

5258 

5340 

5422 

55o3 

5585 

5667 

5748 
6564 

5830 

82 

532 

59I2 

5993 

6075 

61 56 

6238 

6320 

6401 

6483 

6646 

82 

533 

6727 

6509 

6890 

6972 

7053 

7'34 

7216 

7297 

7379 

7460 

81 

534 

7341 

7623 

7704 

7783 

7^66 

7948 

8029 

8110 

8191 

8273 

81 

535 

83  D4 

8435 

85i6 

8397 

8678 

8739 

8841 

8922 

9003 

9084 

81 

536 

9165 

9246 

9327 

9408 

9489 

9370 

9651 

9732 

9813 

9893 

81 

537 

9974 

••55 

•i36 

•217 

•298 

•378 

•439 

•540 

•621 

•702 

81 

538 

730782 

o^63 

0944 

1024 

iio5 

ub6 

1266 

i347 

1428 

i5o8 

81 

539 

1589 

1669 

1700 

1 830 

191 1 

1991  1  2072 

2132 

2233 

23i3 

81 

540 

732394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3o37 

3117 

80 

541 

3.97 

3278 

3358 

3438 

35i8 

3398 

3679 

3739 

3839 

3919 

80 

542 

3999 

4079 

4i6o 

4240 

4320 

4400 

44^0 

4360 

4640 

4720 

80 

543 

4800 

4860 

4960 

5o4o 

5 120 

5200 

5279 

5359 

5439 

55i9 

80 

544 

5599 

5679 

^J-? 

5838 

5918 

5998 

6078 

6157 

6237 

63i7 

80 

545 

6397 

6476 

6006 

6635 

6715 

6795 

6874 

6934  j  7034 

7ii3 

80 

546 

7iy3 

7272 

7352 

743. 

75m 

7590 

7670 

7749   7829 

7908 

79 

547 

79''>7 

8067 

8146 

8223 

83o5 

8384 

8463 

8343 

8622 

8701 

79 

548 

8781 

8860 

8939 

9018 

9097 

9177 

9256 

9335 

9414 

9493 

79 

549 

9,572 

9661 

9731 

9810 

9889 

9968 

••47 

•126 

•205 

•284 

79 

55o 

740363 

0442 

052I 

0600 

0678 

0757 

o836 

09.5 

0994 

1073 

79 

55i 

Il52 

1230 

i3o9 

i3»8 

1467 

1 546 

1624 

1703 

1782 

i860 

79 

552 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2647 

]l 

553 

2723 

2804 

2802 

2961 

3o39 

3ii8 

3196 

3273 

3353 

343I 

554 

35io 

3588 

3667 

3745 

3823 

3902 

39S0 

4038 

4i36 

42i5 

78 

555 

4293 

4371 

4449 

4328 

4606 

4684 

4762 

4840 

491Q 

4997 

78 

..   556 

5075 

5i53 

5231 

5309 

5387 

5465 

5543   5621  1  5699 

7 

5777 

78 

557 

5855 

5933 

601 1 

6089 

6167 

6245 

6323  1  6401 

6479 

6556 

78 

558 

6634 

6712 

6790 

6d68 

6945 

7023 

7101   7179 

7256 

7334 

78 

559 

7412 

7489 

7007 

7643 

7722 

7800 

7878   7933 

8o33 

8110 

78 

56o 

748188 

8266 

8343 

8421 

8498 

8576 

8653   8731 

8808 

8885 

77 

56 1 

8963 

9040 

9118 

9193 

9272 

935o 

9427 

9304 

9582 

9659 

77 

562 

9736 

9814 

9891 

9968 

••43 

•123 

•200 

•277 

•354 

2431 

77 

563 

75o5o8 

o586 

o603 

0740 

08.7 

0894 

0971 

1048 

1125 

1202 

77 

564 

1270 

i356 

/433 

1310 

1 587 

1664 

1741 

1818 

1895 

1972 

77 

565 

20481  2125 

2202 

2279 

2356 

2433 

2309 

2586 

2663 

2740 

77 

566 

2S16;  2893 

2970 

3o47 

3123 

3200 

3277 

3353 

343o 

35o6 

77 

567 

3583  3660   3736 

38i3 

3889 

3966 

4042 

4119 

4195 

4272 

77 

568 

4348 !  4425 

40OI 

4578 

4654 

4730 

4807 

4883 

4960 

5o36 

76 

569 

5ii2  5189 

5265 

5341 

54i7 

5494 

5570 

5646 

5722 

5799 

76 

570 

755875I  5951 

6027 

6io3 

6180 

6256 

6332 

6408 

6484 

6560 

76 

571 

6636;  6712 

6788 

6864 

6940 

7016 

Tst] 

7168 

7244 

7320 

76 

572 

8ih    8230 

7048 

7624 

7700 
8458 

7775 

7927  8oo3 

8079 

76 

573 

83  0^ 

8382 

8533 

8609 

8685  i  8761 

8836 

76 

574 

8912  8988  1  9063  1  9139 

9214 

9290 

9366   9441  ■  9^11 

9392 

76 

57D 

9668^  9743   9^19 

9894 

9970 

••45 

•121   ^196  ^272 

•347 

75 

576 

760422  0498   0073 

0649 

0724 

0799 

0875  0930  !  1025 

IIOI 

75 

577 

1176!  I25i  1  i326 

1402 

1477 

l532 

1627  ,  1702  j  1778 

i853 

75 

578 

1928  2oo3  1  2078 

21 53 

2228 

23o3 

2378  '  2453   2529   2604 

75 

579 

2679  2754   2829 

2904 

2978 

3o53 

3i28  j  32o3  3278 

3353 
9 

75 

N. 

0 

I 

2 

'   1 

4 

5 

6  1  7 

«  1 

10 


A  TABLE   OF   LOGARITHMS   FROM    1    TO   10,000. 


6rfo 

0   1   I 

2 

3 

4 

5 

6 

7 

8     9 

I>. 

76342 >  3Jo3 

3376 

3633 

3727 

36j2 

3^77 

3932 

4027  ,  410, 

~i 

5c>i 

4176  423i 

4326 

4ioo 

4473 

453o 

4624 

4099 

4774  ;  4643 

75 

562 

4923  499^ 

5072 

5i47 

0221 

5296 

5370 

5440 

5o2o  5594 

73 

563 

56^9  5743 

•5si6 

5092 

5935 

6041 

6, ,3 

6190 

6204  ;  6336 

74 

5d4 

6413 

6467 

6562 

6/j35 

67,0 

6763 

6659 

6933 

7007  j  ^(y67 

74 

5b5 

7i56 

7260 

7304 

"379 

7433 

7527 

7601 

7675 

7749  I  7823 

74 

566 

76v6 

797  a 

8046 

8120 

6194 

6263 

8342 

84.6 

8490  i  85o4 

74 

567 

8633 

6712 

6766 

8^0o 

8934 

90o3 

9032 

9106 

9260  93o3 

74 

066 

9377 

9-.3I 

9323 

9399 

9'J73 

9746 

9820 

9)^94 

9^06  j  ••42 

74 

569 

770113 

0109 

02()3 

o33c) 

04,0 

0484 

0007 

o63i 

0705  1  0778 

74 

590 

770352 

0926 

0999 

1073 

1.46 

1220 

1293 

1367 

1440  ;  i5i4 

"^i 

5^. 

1 567 

IbJi 

173'. 

i6o3 

1661 

1955 

2026 

2, 02 

2175  ■  2248 

"?? 

5g2 

2322 

2393 

2  463 

2342 

26,5 

2668 

2762 

2  :35 

2906   29^1 

'^l 

5^3 

3o5J 

3126 

3201 

3274 

3348 

3421 

3494 

3567 

3040 

3713 

7? 

^4 

3766 

3o5a 

3933 

400') 

4079 

4,32 

4223 

4296 

4371 

4444 

7? 

5^5 

45.7 

4093 

4663 

4736 

4S09 

4-V)2 

4935 

5o26 

D,00 

5,73 

7^ 

5^ 

5246 

5319 

5392 

5465 

5536 

56,0 

5o63 

5756 

5-.29  1  5902 

^i 

597 

5974  6047 

6120 

6193 

6263 

6333 

641, 

6463 

6530 

6029 

f^. 

09:! 

6701 

6774 

6M6 

6919 

6992 

7064 

7137 

7209 

7202 

7334 

73 

599 

7427 

7^99 

7572 

7'J44 

77'7 

77^ 

7662 

7934 

8000 

8079 

72 

600 

778i5i 

8224 

8296 

836<3 

8441 

85,3 

8585 

8633 

8730 

8802 

72 

601 

8-74 

8947 

9019 

9091 

9163 

9236 

9306 

93So 

^32 

9524 

72 

602 

9596 

9JJ9 

9741 

9S13 

9563 

9957 

••29 

•101 

•.73 

•243 

72 

6o3 

780317 

o3d9 

0461 

o333 

o6o3 

0677 

0749 

0821 

0693 

0965 

72 

604 

io37 

1109 

1181 

1253 

,324 

1396 

1466 

1540 

16,2 

1084 

72 

6o5 

1755 

1627 

1S99 

1971 

2042 

21,4 

2,86 

2238 

2629 

2401 

72 

606 

2473  2344 

2616 

i-o6i 

2739 

2^3l 

2902 

2974 

3j40 

3,17 

72 

607 

3169 

3  200 

3332 

3403 

3475 

3546 

36i8  !  36S9 

3761 

3632 

7» 

6.6 

3904 

3975 

4046 

41.8 

4189 

426, 

4332 

44o3 

4475 

4546 

7» 

609 

4017 

4039 

4760 

463, 

49^2 

4974 

5o45 

5i,6 

5i67 

52:9 

7» 

610 

785330 

5401 

5472 

5543 

56,5 

56-86 

5757 

5828 

5699 

5970 

7» 

611 

604. 

6112 

6i83 

6234 

6325 

6396 

6467 

6538 

6009 

6060 

7» 

6i2 

6751 

6622 

6-!93 

6964 

7033 

7.06 

7177 

7243 

73,9 

7390 

7* 

6i3 

7460 

7331 

7602 

76j3 

7744 

7813 

7683 

7956 

8027 

8096 

7» 

614 

816S 

6239 

83io 

83^. 

845, 

8322 

8093 

8663 

8734 

8804 

7» 

6.5 

8-i75 

8946 

9016 

9037 

9,37 

9220 

9299 

9359 

9440 

95,0 

7» 

616 

9561 

96  J 1 

9722 

9792 

9363 

9933 

•••4 

••74 

•144 

•2,5 

70 

617 

790235;  o3j6 

0426 

o49^J 

0307 

0037 

0707 

0778 

0646 

0918 

70 

616 

093s  10J9 

1129 

"97 

,269 

i340 

1410 

,430 

,5  JO 

1620 

70 

619 

1091  17^1 

i83i 

190, 

1971 

»■" 

2111 

2 181 

2202 

2322 

70 

620 

792392  2462 

2  332 

2602 

2672 

2742 

28,2 

2S82 

2932 

3022 

70 

621 

3092  3 102 

323i 

33ji 

3371 

3441 

33,, 

3561 

3ooi 

372, 

70 

622 

3793  3660 

393J 

4joo 

4070 

4139 

4209 

4279 

43.49 

4418 

70 

623 

44^3  4556 

4627 

4^>97 

4767 

4836 

4906 

49-6 

5043 

5ii5 

70 

624 

5i65  5234 

5324 

53)3 

5463 

5532 

56o2 

5672 

5741 

56ii 

70 

620 

5660  59  ;9 

6019 

60.66 

6,36 

6227 

6297 

6366 

6436 

65o5 

69 

626 

6574  60^4 

6713 

6762 

6:132 

692, 

6990 

7060 

7.29 

7198 

69 

627 

7i63  73J7 

7406 

7473 

-545 

7614 

7063 

7732 

762, 

7690 

69 

626 

79^0  6029 

8.93 

8,67 
8658 

6236 

83o3 

8374 

8443 

63,3 

85^2 

69 

629 

863i  8720 

8739 

8927 

8996 

9065 

9,34 

9203 

9272 

69 

63o 

799341'  9409 

9478 

9547 

9616 

9635 

9754 

9S23 

9892 

9961 

69 

63 1 

800029  0096 

0167 

03)6 

o3o5 

0373 

0442 

o3,i 

0060  1  O^i^-i 

69 

632 

0717  0766 

oS34 

0923 

0992 

1061 

>'29 

1.98 

1266 

i335 

69 

633 

1404  1472 

i54i 

1009 

1676 

«747 

l8lD 

1684 

,932 

2021 

69 

634 

2069  2 1 56 

2225 

2293 

2333 

2432 

23oo 

2368 

2037 

2735 

^ 

635 

2774,  2642 

2910 

2979 

3o47 

3i,6 

3,84 

3232 

332, 

3369 

636 

3i57 

3525 

3394 

3:j62 

3730 

37<^8 

3S67 

3933 

4oy3 

407, 

68 

1  637 

4139 

42  oS 

4276 

4344 

4412 

4430 

4548 

4616 

46:53 

4733 

63 

636 

432, 

4339 

4937 

5o25 

50)3 

5,61 

5229 

5297 

5365 

5433 

68 

639 

55oi|  5569 

5637. 

5705 

5773 

5641 

3905 

5976 

6044 

6, ,2 

68 

N. 

0   1   I 

2 

"^ 

^ 

5 

6 

7 

8   i   9 

D. 

A  TABLE   OF   LOGARITHMS   FROM    1    TO    10,000. 


11 


N. 

0 

I 

2 

3415     6!7:BJ9 

JD. 

640 

806180 

6248 

63i6 

6384 

1  6451 

65 19  6387  :  6655   6723  1  6790 

68 

641 

6858 

6926 

6994 

706. 

71^9 

7'97 

:  7264  ,  7332  7400  1  7467 

68 

642 

7535 

7003 

i:^^ 

7738 

7806 

1  7873 

i  7941  1  8008   8076  ;  8143 

68 

643 

8211 

8279 

8414 

8481 

8349 

:  8616  1  8684   8731  i  8818 

67 

644 

8886 

8953 

9021 

9088 

9i56 

9223 

,  9290  1  9338   9425  1  9402 

67 

1  645 

9560 

9627 

9694 

9762 

9829 

9896 

\   9964 

••3 1   ••98 

•l65 

!  67 

1  646 

810233 

o3oo 

o3o7 

0434 

o5oi 

o569 

0636 

0703   0770 

0837  1  67 

1  647 

0904 
1575 

0971 

1039 

1106 

1173 

1240 

1  1 307 

1374 

i44« 

i5o8 

67 

648 

1642 

1709 

1776 

1843 

1^10 

'  1977  1  2044 

2111 

2178 

67 

649 

2245 

23l2 

2379 

2445 

25l2 

2379 

2646 

2713 

2780 

2847 

67 

65o 

812913 

2980 

3047 

3114 

3i8i 

3247 

!33.4 

338i 

3448 

35i4 

'   67 

65i 

358i 

3648 

3714 

37^1 

3848 

3914 

3981 

4048 

4H4 

4181 

1   67 

652 

4248 

43 14 

4381 

4447 

4514 

4581 

4047 

4714 

47S0 

4847 

i   67 

653 

49'3 

4980 

5046 

5ii3 

3179 

5246 

53i2 

5378 

5445 

531, 

66 

654 

5578 

5644 

57.1 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

66 

655 

6241 

63o8 

6374 

6440 

65o6 

6373 

6639 

6703 

6771 

6cS38 

66 

656 

6904 

6970 

7036 

7102 

7./.9 

7235 

7301 

7367 

7433 

7499 

66 

657 

7565 

703 1 

7698 

7764 

783o 

7896 

7962 

8028 

8094 

8160 

66 

658 

8226 

8292 

8358 

8424 

8490 

8536 

8622 

8668 

8734 

8820 

66 

659 

8885 

8931 

9017 

9003 

9149 

92i5 

9281 

9346 

9412 

947B 

66 

660 

819544 

9610 

9676 

9741 

9807 

9873 

9939 

•••4 

••70 

•i36 

66 

661 

820201 

0257 

o3J3 

0399 

0464 

o53o 

0393 

0661 

0727 

0792 

66 

662 

0858 

0924 

09H9 

103J 

1120 

1186 

|2JI 

i3i7 

l3S2 

14-4'^ 

66 

663 

i5i4 

1379 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2io3 

65 

664 

2168 

2233 

2299 

2304 

243o 

2493 

2j6o 

2626 

2691 

2756 

65 

665 

2822 

2887 

2932 

3018 

3o83 

3i48 

32i3  i  3279 

3344 

3409 

65 

666 

3474 

3539 

36o5 

3670 

3735 

3boo 

3665  i  3980 

3996 

4061 

63 

667 

4126 

4191 

4256 

4321 

4386 

445 1 

4^16  1  4381 

4646 

47  n 

65 

668 

4776 

4«4i 

4906 

4971 

5o36 

3101 

5i66  i  523i 

5296 

536i 

65 

669 

5426 

5491 

5556 

5021 

56b6 

575. 

58i5  1  5860 

5945 

6010 

65 

670 

826075 

6140 

6204 

6269 

6334 

6399 

6464  j  6528 

6393 

6658 

65 

671 

6723 

6787 

6832 

6917 

6981 

7040 

7111 

7175 

7240 

73o5 

65 

672 

7369 

7434 

7499 

7:^03 

7628 

7692 

77:>7 

7821 

7.S.S6 

7931 
839^ 

65 

673 

8oi5 

8080 

8144 

8209 

8273 

8338 

8402  8467 

853 1 

64 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

92  J9 

64 

675 

9304 

936^ 

9432 

9497 

9561 

9625 

9690 

97^4 

9818 

eS.-:2 

64 

676 

9947 

**i  1 

..75 

•1J9 

•204 

•208 

•3J2 

•396 

•460 

'325 

64 

^n 

83ood9 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

678 

I23o 

1294 

i358 

1422 

i486 

i35o 

1614 

1678 

1742 

1806 

6^ 

679 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

238i 

2445 

64 

680 

832509 

2573 

2637 

2700 

2764 

2828 

2892 

2950 

3020 

3oS3 

64 

681 

3.47 

3211 

3273 

3J38 

3402 

3466 

3330 

339J 

3657 

3721 

64 

682 

3784 

3d48 

39.2 

3975 

4039 

4io3 

4106 

423o 

4294 

4337 

64 

683 

4421 

4484 

4348 

4011 

4673 

4739 

4802 

4866 

4929 

4993 

64 

684 

5o56 

5l20 

5i83 

5247 

53io  1 

5373 

5437 

55oo 

5564 

5627 

63 

685 

569. 

5754 

5817 

58di 

5944  1 

6007 

6071 

6134 

6197 

6261 

63 

686 

6324 

6357 

6431 

6314 

6377  1  6641 

6734  :  6767 

663o 

6894 

63 

687 

6957 

7020 

7083 

7i-*6 

7210  j  7273 

7336  ;  7399 

7462 

7323 

63 

688 

7388 

7632 

77i5 

111^ 

7841  ;  7904 

7967  •  8o3o 

8093 

81/-6 

63 

689 

8219 

8282 

8345 

8408 

847 «  i  8534 

8397   8660 

8723  1 

8786 

63 

690 

838849 

89.2 

8975 

9o38 

9101  1  9164 

9227  '■  9289 

9312 : 

9413 

63 

691 

9478 

9341 

9604  9(^bi 

9729  j  9792 

9833  991a 

99^'   ••43 

63 

692 

840106 

0169  1  0232  '  0294 

0357  1  0420  1 

0482   0345 

0608   0671 

63 

.693 
694 
693 

0733 

0796  0859 

0921 

0984  1  1046  1 

1109  •  1172 

1234   1297 

63 

'^^9 

1422   1485 

1347 

1610  1  1672  I 

1733  i  1797 

i860   1922 

63 

1983 

2047   2110 

2172 

2235  ;  2297  j 

2360  i  2422 

2484  1  2347 

62 

696 

2609 

2672   2734 

2796 

2839   2921  I 

2983  .  3046 

3io8  1  3170 

62 

697 

3233 

3295   3357 

3420 

3482  j  3544  j 

36o6   3669 

373 1  i  3793 

62 

698 

3855 

3918  3980 

4042 

4104  j  4166  . 

4229   4291 

4353   4413  j 

62 

699 

4477, 

4339  ^  4601 

4664 

4726   4788  ■ 

485o   4912 

4974   5o36  1 

1 

62 

N. 

«  1 

I     2 

3 

4  j   5   , 

6   i   , 

8  1  9  1 

D. 

12 


A  TABLE   OF   LOGARITHMS   FROM   1    TO   10,000. 


1  X. 

i    °   ,    '       M   M   ^ 

1  ^  L^_ 

7 

8   i  9 

D. 

700 

840098!  5 1 60 

:  5522 

52 S4  i  5346 

1  5408  5470 

5532 

5594  :  5656 

62 

701 

5718 

I  5780 

,  5842 

5904 

5966 

1  6028  6090 

6i5i 

62i3  6275 

62 

702 

6337 

1  6399 

j  6461 

6523 

6385 

;  6646  1  6708 

6770 

6832  6894 

62 

703 

6955 

1  70'7 

7079 

7141 

7202 

'  7264 

7326 

7388  I  7449  75 i  I 

62 

704 

7573 

1  7634 

Itf. 

7758 

7819 

'  7^:^81 

7943 

8004   8066   8128 

62 

705 

8189 

1  825i 

«374 

8435 

8497 

8539   8620  1  8682  :  8743 

62 

706 

88o5 

;  8866 

8928 

89S9 

905. 

i  9"2 

j  9174  ;  9235   9297  :  9358 

61 

707 

9419 

9481 

9^42 

9604 

9663  :  9726 

i  97^8  !  9849  99' »  ;  9972 

61 

708 

85oo33 

0095 

1  01 56 

0217 

0279  !  o34o 

1  0401   0462   0324  1  o585 

61 

709 

0646 

0707 

1  0769 

o83o 

089  r 

0932 

1  1014   1075   ii36  1  1197 

61 

710 

851258 

1 3  20 

1  i3S. 

1442 

i5o3 

1 564 

:  1625   1686   1747  '  1809 

61 

711 

1870 

1931 

!  1992 

2o53 

2114 

2173 

2236  i  2297   2358  ;  2419 

61 

712 

2480 

2041 

1  2602 

2663 

2724 

2785 

:  2846  !  2907  2968  i  3029 

61 

7.3 

3090 

3i5o 

i  32II 

3272 

3333 

3394 

3455  1  3316  3377  '  3637 

61 

714 

3698 

3739 

!  3cJ2o 

3881 

3941 

4002 

'  4o63"  1  4124  41 85  .  4243 

61 

7.5 

43o6 

4367 

'   4428 

44S8 

4549 

4610 

4670  ;  4731 

4792   4852 

61 

716 

4913 

4974 

'  5o34 

5095 

5i56 

52i6 

5277  ;  5337 

5398  ■■   5459 

61 

717 

5519 

5580 

5640 

5701 

5761 

5823 

,5882  i  5943 

6oo3  6064 

61 

71S 

6124 

6i85 

6245 

63o6 

6366 

6427 

6487  ■■   6348 

6608  !  6663 

6o 

719 

6729 

6789 

6s5o 

6910 

6970 

7o3. 

7091  1  71 52 

7212   7272 

60 

720 

857332 

7393 

l^'-^ 

75.3 

7^74 

7634 

7694  i  7755 

73i5  7875 

60 

1  72. 

7935 

7993 

8od6 

8116 

8176 

8236 

8297   8357 

8417  1  8477 

60 

722 

8537 

8097 

8637 

8718' 

8778 

8838 

8898   8958 
9499  9559 

9018  1  9078 

60 

723 

9i38 

9.98 

9258 

93.8 

9379 

9439 

9619  !  9679 

60 

724 

9789 

9799 

9859 

9918 

9978 

••38 

••98   •i58 

•218  ;  ^278 

60 

725 

86o338 

039^ 

0408 

ODl8 

0378 

0637 

0697   0737 

0817  '   0877 

6a 

726 

oq37 

0996 

io56 

1II6 

1176 

1236 

1293   1 355 

I4i5  ■  1473 

60 

727 

1 534 

1594 

1 654 

1714 

1773 

i833 

1893   1932 

2012   2072 

60 

728 

2l3l 

2191 

225l 

23lO 

2370 

243o 

2489   2349 

2608  !  2668 

60 

729 

2728 

2787 

2847 

2906 

2966 

3o25 

3o83  3 144 

3204  3263 

60 

730 

863323 

33S2 

3442 

35oi 

356i 

3620 

368o  3789 

3799  3858 
4392  ;  4452 

59 

73. 

3917 

3977 

4o36 

4096 

4i55 

4214 

4274  4333 

59 

732 

4511 

43'70 

463o 

4689 

4748 

4-^08 

4867   4926 

4q85   5o45 

59 

733 

5io4 

5i63 

5222 

5282 

5341 

5400 

5459   5319 

5578   5637 

59 

734 

5696 

5755 

5ai4 

5874 

5933 

5992 

6o3i   6iio  6169  6228 

59 

735 

62^7 

6346 

6405 

6465 

6324 

6583 

6642  6701  !  6760  6819 

59 

736 

6878 

6937 

6996 

7o55 

7i«4 

7173  :  72321  7291  j  7350  7409 

59 

7^7 

7467 

7526 

708D 

7644 

7703 

7-:62 

7821   7.^80  :  7939   7998 
8409  8468  I  8327   85 S6 

59 

738 

8o56 

8ii5 

8174 

8233 

8292 

83  5o 

59 

739 

8644:  8703 

8762  1  8821 

8879  1  8938 

8997   9o56  9114  ;  9173 

59 

740 

869232'  9290 

9349 

9408 

9466  ;  9525 

9584  9642  9701  1  9760 

59 

74" 

981^  9877 
870404  0462 

993J 

9994 

••53 

•ill 

•170   ©228  i  •287   ^345 

It 

742 

O02I 

0579 

o638 

0696  '  0755  081 3  :  0872  '  0930 

743 

0989  1047 

II06 

1164 

1223 

1281 

i339   1898  :  1456  :  i5i5 
1928   1 98 1  •  2040   2098 

58 

744 

1573  i63i 

1690  1 

1748 

1806 

|865 

58 

743 

2 1 56  221 5 

2273 

233 1 

23^9 

2448 

2 306   2564   2622   26S1 

58 

746 

2-'39  27Q7 

2855 

2913 

2972 

3o3o   3o88  3146  i  3204  3262 

58 

747 

3321 

3379 

3437 

3495 

3553   36 n 

3669   2727   3785   3844 

58 

748 

3902 

3960 

4018  ;  4076 

4i34  ■■   4192 

4230  ,  43o8  ,  4366  ,  4424 

58 

749 

4482 

4540 

4598  1  4656 

47»4  :  4772 

4830  4888  !  4945  '  5oo3 

58 

75o 

875061 

569^ 

5177  1  5235 

5293  ;  535i  :  5409   5466   5524   55«2  ' 

58 

731 

5640 

5756   58 1 3 

5871  :  5929 

5987  6045  6102  6160  : 

53 

702 

6218 

6216 

6333  ,  6391 

6449  ;  65o7 

6564  6622   6680  6737  1 

58 

753 

6-95  6853 

6910  6968 

7026  i 

70S3  ;  7MI   7'99   7256   7314 

58 

754 

7371  7429 

7487 

7344  { 

7602 

7659  :  nn    7774  7832  7889  1 

58 

755 

7947 

8004  i  8062 

8119  1 

S177  • 

8234  ■  8292   8349  8407   8464  i 

57 

756 

8522 

8579  1  8637 

8694! 

8732  : 

8809   8866   8924   8981   9039  ; 

57 

757 

9096 

9i53   9211 

9268  ' 

9325  93 -(3   9-140   9497   9333  9612  ^ 

37 

nbS 

9669  972£ 

9784 

9841  ; 

9893  ;  99 ')6   ••i3   ^^70  ^127   •183 

V 

759 

880242  0299 

o356 

04i3  0471   0028   o585   0642   0699   0756 

57 

N. 

0  1   I   i   2     3   i  4  ,   5   ,  6     7     8   :   9 

D. 

A   TABLE    OF   LOGARITHMS   FKOM   1    TO    10.000. 


18 


Hn. 

0 

I   1   2 

3 

4 

5 

6 

7 

3 

9 

D.  1 

760 

880814 

0871 

0928 

0985 

1042 

1099 

ii56 

12.3 

1271 

i328 

57; 

76, 

1 385 

1442 

1499 

1 556 

i6i3 

1670 

1727 

1784 

1841 

1898 

57 

762 

1955 

2025 

2012 

2069 

2126 

2 1 83 

2240 

m 

2354 

241 1 

2468 

571 

763 

258i 

2638 

2695 

2752 

2809 

2923 

2980 
3548 

3o37 

57 1 

764 

3093 

3i5o 

3207 

3264 

332. 

3377 

3434 

3491 

36o5 

^7i 

765 

366 1 

37.8 

3775 

3832 

3888 

3945 

4002 

4039 
4625 

4n5 

4172 

57 

766 

4229 
4795 

4285 

4342 

4399 

4455 

43.2 

4569 

4682 

4739 

^7 

767 

4852 

4909 

it. 

5o22 

5078 

Dl33 

5192 

5248 

5303 

57 

768 

536, 

5418 

5474 

5587 

5644 

5700 

6321 

58i3 

5870 

11 

769 

5926 

5983 

6039 

6096 

6i52 

6209 

6265 

6378 

6434 

770 

88649. 
7034 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

56 

771 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

73o5 

7361 

56 

772 

7617 

7(^74 

773o 

7786 

7842 

7898 

7935 

8011 

8067 

8.23 

56 

773 

8179 

8236 

8292 

8348 

8404 

8460 

85.6 

8573 

8629 

8685 

56 

774 

874. 

liu 

8853 

8909 

8965 

9021 

9077 

9134 

9.90 

9246 

56 

775 

9.302 

9414 

9470 

9526 

95S2 

9638 

9694 

9730 

9S06 

56 

776 

9862 

9918 

9974 

••30 

••86 

•141 

•'97 
0756 

•233 

•309 

•365 

56 

777 

890421 

0477 

o533 

o589 

0645 

0700 

0812 

0868 

0924 

56 

77« 

a1 

io35 

109. 

1147 

I203 

1259 

i3i4 

1370 

1426 

1482 

56 

779 

1593 

.649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

56 

780 

892095 

2i5o 

2206 

2262 

23.7 

2373 

2429 

2484 

2540 

2395 

56 

781 

265. 

2707 

2762 

2818 

2873 

2929 

2985 

3o4o 

3096 

3.5. 

56 

782 

3207 

3262 

33.8 

3373 

3429 

3484 

3540 

3595 

365. 

3706 

56 

783 

3762 

38 1 7 

3873 

3928 

3984 

4039 

4094 

4i5o 

42o5 

4261 

55 

7S4 

43 16 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

48.4 

55 

785 

4870 

4925 

49S0 

5o36 

5091 

5.46 

5201 

5257 

53.2 

5367 

55 

786 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

55 

787 

5975 

6326 

6o3o 

6o35 

6140 

6.95 

625i 

63o6 

636i 

64.6 

6471 

55; 

788 

658. 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

55 

789 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7317 

7572 

55 

790 

897627 

7682 

7737 
8286 

7792 

7847 

7902 

7957 

80.2 

8067 

8.22 

55 

79' 

8.76 

823. 

8341 

8396 

843. 

8306 

856i 

86.5 

8670 

55 

J92 

8725  8780 

8835 

8890 

8944 

8999 

9054 

9109 

9«64 

92.8 

55' 

?93 

9273 

9328 

9383 

9437 

9492 

9^47 

9602 

9656 

97.1 

9766 

55 1 

?94 

9821 

9875 

9930 

99SD 

••39 

••94 

•149 

•203 

.•258 

•3.2 

ii\ 

195 

900367 

0422 

0476 

o53i 

o586 

0640 

0695 

0749 

^0804 

0859 

551 

,96 

091 3 i  0968 

1022 

1077 

1.3. 

1186 

1240 

1295 

1349 

1404 

55; 

/97 

1458!  i5i3 

1 567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

^/l 

;98 

2003!  2057 

21.2 

2.66 

222. 

2273 

2329 
2873 

2384 

2438 

2492 
3o36 

54 

799 

2547I  2601 

2655 

2710 

2764 

2818 

2927 

2981 

54 

^00 

903090'  3 1 44 

3.99 

3253 

3307 

3361 

34.6 

3470 

3524 

3578 

54 

Boi 

3633  3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

541 

802 

4174'  4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

466. 

i^\ 

So3 

4716,  4770 

4824 

4878 

4932 

4986 

5o4o 

5094 

5.48 

5202 

54 ! 

804 

5256^  53.0 

5364 

54.8 

5472 

5326 

558o 

5634 

5688 

5742 

541 

&o5 

5796'  585o 

5904 

5958 

6012 

6066 

61.9 

6173 

6227 

6281 

54 

3o6 

6335  6389 

6443 

6497 

655 1 

66o4 

6658 

6712 

6766 

6820 

54! 

807 

6874:  6927 

6981 

7035 

7089 

7143 

7196 

7250 

73o4 

7358 

54  i 

808 

7411 

7465 

m 

7573 
8110 

7626 

7680 

7734 

7787 

7841 
8378 

S? 

^A 

809 

7949 

8002 

8.63 

8217 

8270 

8324 

54 

810 

908485 

8539 

8592 

8646 

It?. 

8753 

8807 

8860 

8914 

8967 

'A 

811 

9021 

9074 

9128 

9.81 

9289 

9342 

9396 

9449 

95o3 

54 

812 

9556 

9610 

9663 

97.6 

9770 

9823 

9877 

9930 

9984 

••37 

53 

8i3 

9 1 009 1 

0144 

0197 

025l 

o3o4 

0358 

04.1 

0464 

o5i8 

0371 

53 

814 

0624 

0678 

073; 

0784 

o838 

0891 

0944 

0998 
i33o 

io5i 

1.04 

53 

8i5 

1.58  .2. 1 

1264 

i3i7 

1371 

1424 

1477 

1 584 

i637 

53 

816 

1690  .743 

1797 

i85o 

1903 

1936 

2009 

2o63   21.6 

2.69 

53 

817 

2222  2275 

2328 

23Si 

2435 

2488 

254. 

2594   2647 

2700 

53 

818 

27531  2806 

2859 

29.3 

2966 

30.9 

3072 

3.25 

3.78 

323i 

53 

8.9 

3284,  3337 

3390 

3443 

3496 

3549 

36o2 

3655 

3708 

3761 

53 

N. 

0   1   1 

2 

3 

4 

5 

6 

7 

8 

9 

D. 

14 


A   TABLE   OF   LOGARITII.MS   FROM    1    TO    10.000. 


N. 

1    0    !    I       2 

1  ^ 

1  ' 

3   1   6   !   7 

8  :  9 

D. 

820 

913-14  3S67    3920 

3973 

4026 

4079 

4i32  1  4184 

4237^,  4290 

53 

821 

4343  4396   4449 

45o2 

4555 

4608 

4660 

47'3 

4766   4b 19 

53 

i  822 

4872  49->5  4977 
3400  5453   55o5 

5o3o 

5oN3 

5i36 

5,89 

5241 

5294  1  5347 

5a 

i  82J 

5558 

56ii 

5664 

5716 

5769 

5b2  2  !  5b75 

53 

'  824 

5927  59S0  6o33 

6o83 

61 3S 

6191  j  6243 

6296 

6349  i  ^401 

53 

:  82J 

6-,J4  65o7  '  6559 

6612 

6664 

:  6717   6770 

6822 

6873  ;  6927 

53 

i  826 

1  6960  7033   7085 

7138 

7190 

1  7243   7293 

7348 

7400  j  7453 

53 

;  «27 

!  7OUO  7558   7611 

7663 

7716 

7768  1  7ti2o 

7873 

7923   7978 

52 

828 

I  bo3o  8o83   8135 

8188 

8240 

1  8293   8345 

8397 

845o   8302 

52 

■  829 

8555  8607  ,  8609 

87.2 

8764 

8816   8869 

8921 

8973   9026 

52 

1  83o 

919078  9i3o  9183 

9235 

9287 

9340  9392 

9444 

9496 

9549 

32 

j  83, 

9601  9653   9706 

9738 

9810 

9862   9914 

9967 

••,9 

••71 

52 

832 

92JI23  0176   0228 

o2i;o 

o332 

o384 

1  0436 

0489 

o54i 

0593 

52 

833 

0045  0697   0749 

080 1 

o853 

0906 

0958 

10.0 

1062 

1114 

52 

834 

1166  1218  ;  1270 

l322 

1374 

1426 

1478 

i53o 

i582 

1 634 

52 

835 

i686  1733  ,  1790 

1842 

1894 

1946 

1998 

2o5o 

2.02 

2.54 

02' 

836 

2206  2258  i  23lO 

2362 

2414 

2466 

25.8 

2570 

2622 

2674 

52 

837 

2725  2777 

2829 

2b8i 

29J3 

2q85 

3o37 

3089 

3.40 

3192 

52, 

83ii 

3244  3296 

3348 

3399 

345 1 

33o3 

3555 

3607 

3658 

3710 

52 

839 

3702  38i4 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

52 

1  840 

924279:  433i 

4383 

4434 

44'^6 

4538 

4589 

4641 

4693 

4744 

52 

841 

-479^  4848 

4899 

4951 

5oo3 

5o54 

5io6 

5,57 

5209 

5261 

52 

842 

53 12,  5364 

540 

5467 

55i8 

5570 

5621 

5673 

5725 

5776 

52 

84  i 

5o2b  5879 

5931 

5982 

6o34 

6o85 

6137 

6.88 

6240  !  6291 

5i 

844 

63^21  6394 

6443 

6497 

6543 

6600 

665 1 

6702  i  6754  i  68o3 

5i 

845 

6857  6908 

6959 

701 1 

7062 

7114 

7163 

7216   7268 

73.9 

5i 

840 

7370  7422 

7473 

7324 

7576 

7627 

7678 

7730  1  7781 
8242  ;  8293 

7832 

5i 

847 

7bb3  7935 

7986 

8o37 

8ob8 

8140 

9191 

8703 

8343 

'5i 

848 

8396  8447 

8498 

8549 

8601 

8652 

8754  1  8803 

8857 

5i 

849 

890b  8939 

9010 

9061 

9112 

9i63 

9215 

9266   93.7 

9368 

5i 

85o 

929419  9470 

9521 

9372 

9623 

9674 

9725 

9776  1  9827 

9879 

5i 

85 1 

9930  9981  j  ••32 

••tj3 

•i34 

•i85 

•236 

•287  1  •338  !  •389 

5i 

852 

930^40 i  0491 

o542 

0592 

0643 

0694 

0745 

0796  j  0847   0898 

5i 

853 

0949  1000 

io5i 

1102 

n53 

1204 

.254 

i3o5   i356   1407 

5, 

854 

1 45b  i5o9 

i56o 

1610 

,66i 

1712 

1763 

1814   iti65  1  1915 

5,1 

855 

1966  2017 

2068 

2118 

2169 

2220 

2271 

2322  1  2372  I  2423 

5i 

856 

2474  2024 

2573 

2626 

2677 

2727 

2778 

2829  1  2b79  1  2930 

5i 

857 

29b I  3o3i 

3o82 

3!33 

3ib3 

3234 

3285 

3335  j  3386 

3437 

5i 

858 

34b7  3533 

3589 

3639 

3690 

3740 

3791 

3841  1  3892 

3943 

5.1 

809 

3993  4044 

4094 

4143 

4195 

4246 

4296 

4347   4397 

4448 

5.' 

860 

93449*^  4549 

4099 

465o 

4700 

475. 

4B01 

4852   4902 

4953 

5o; 

861 

5oo3  5o54 

5io4 

5i54 

5203 

3255 

53o6 

5356  1  5406 

5457 

5oi 

862 

5J07  5558 

56o8 

5658  , 

5709 

5759 

5809 

5860  i  59.0 

5960 

5o 

863 

6011  6061 

6111 

6162 

6212 

6262 

63.3 

6363  i  6413 

6463 

5o 

864 

65 1 4  6064 

6614 

6665  1 

6715 

6763 

68i5 

6865  1  6916 

6966 

5o 

865 

7016  7066 

7117 

7167  1 

7217 

7267 

73,7 

7367  1  7418 

7468 

5o 

866 

7»b  7563 

7618 

^668  ; 

7718 

7769 

78.9 

7869  i  7919 

7969 

8470 

5o 

867 

8019  8069 

81 19 

8169  -■ 

8219. 

8269 

8320 

8370  j  8420 

5oi 

86:i  1 

8520  8570 

8620 

8670 

8720 

8770 

S820 

8870  8920   8970 

5o] 

869' 

9020  9070 

9120 

9170  ^ 

9220 

9270 

9320 

9369  I  9419   9469 

5o! 

870 

939319  9369 

9619 

9669  i 

97«9 
0218 

9769 

98,9 

9869  i  99.8  9968 

5o 

871 

940018  0068 

0118 

0168 

0267 

o3i7 
o8i5 

o367   0417  i  0467 

5o 

872 

03 1 6  o566 

0616 

0666  • 

0716 

0765 

o865  1  0913  j  0964 

5oi 

873 

1014  1064 

1114 

ii63 

12l3 

1263 

i3.3 

i362  i  1412  !  1462 

5oi 

^''^ 

i5ii  i56i 

1611 

1660 

1710 

1760 

1809 

1859  i  1909  1  1938 

5o 

873 

2008,  2038 

2107 

2157 

2207 

2236 

23o6   2355  '   2403  i  2455 

5o 

876 

25o4  2554 

26o3 

2653 

2702 

2752 

2801   285i  ;  2901  i  2930 
3297   3346  :  3396  !  3445  , 

5o 

877 
878 

3ooo  3049 

3099  3148 
3393  j  3643 

3.98 

3247 

59  • 

3493  3544 

3692 

3742 

3791   3841  :  3H90  i  3939  1 
4285   4335  :  4384  ;  4433  ' 

09, 

879 

39^9  4038 

4o«8  4137 

4186 

4236 

591 

N. 

0     1   1  2  i  3 

4 

5 

6   1   7   1   8   i   9   j 

^-J 

A  TABLE   OF   LOGARITHMS   YUOM   1    TO    10,000. 


15 


N. 

0 

I       2 

3 

4 

5 

6 

7 

8 

9 

D. 

88o 

944483 

4532   458 I 

463 1 

4680 

4729 

4779 

4828 

iV'' 

4927 

49 

88 1 

4976 

5o25   5074 

5i24 

5.73 

5222 

5272 

532. 

5370 

54.9 

49 

882 

5469 

55 1 8  5567 

56i6 

5665 

57.5 

5764 

58.3 

5862 

39.2 

49 

883 

5961 

6010  60^9 

6108 

6.57 

6207 

6236 

63o3 

6354 

6403 

49 

884 

6452 

65oi   655i 

6600 

6649 

6698 

6747 

6796 

6845 

6^94 

.49 

885 

6943 

6992  1  7041 

7090 

7.40 

7.89 

7238 

7287 

7336 

7383 

49 

886 

7434 

7483   7532 

758 1 

7630 
8.19 

7679 

7728 

Vl 

7826 

7873 

49 

887 

7924 

7973  1  8022 

8070 

8.68 

8217 

8266 

83.5 

8364 

49 

888 

84i3 

8462  ;  85ii 

856o 

8609 

8657 

8706 

8755 

8804 

8853 

49 

889 

8902 

8951  i  8999 

9048 

9097 

9.46 

9195 

9244 

9292 

934. 

49 

890 

949390 

9439  9488 

9536 

9585 

9634 

9683 

973 1 

9780 

9829 

49 

891 

9,^78 

9926  1  9975 

••24 

••73 

•12. 

•.70 

•2.9 

•267 

•3.6 

49 

892 

95o365 

0414 ;  0462 

o5ii 

o56o 

0608 

0657 

0706 

0734 

oSo3 

49 

893 

oaoi 

0900  i  0949 

0997 

1046 

1095 

1.43 

1.92 

1240 

1289 

49 

894 

895 

1338 

i386   I435' 

1483 

1 532 

i5bo 

1629 

1677 

1726 

1773 

11 

1823 

1872  1  1920 

1969 

20.7 

2066 

21.4 

2.63 

22H 

2260 

896 
897 

23oy 

2356  '  24o5 

2453 

2502 

255o 

lif, 

2647 

2696 

2744 

48 

2792 

2841  i  28S9 

2938 

2986 

3o34 

3.3. 

3i8o 

3228 

48 

898 

3276 

3325  1  3373 

342, 

3470 

35.8 

3566 

36i5 

3663 

3i.i 

48 

8?9 

3760 

3!:lo8   3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48 

900 
901 

954243 

4291  1  4339 

4387 

4435 

4484 

4532 

458o 

4628 

4677 

48 

4725 

4773  i  4821 

4869 

4918 

4966 

5o.4 

5o62 

5.10 

3138 

48 

902 
903 
904 
905 

6207 

5255  i  53o3 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

5688 

5736  i  5784 

5832 

5880 

5928 

5976 

6024 

6072 

6.20 

48 

6168 

6216   6265 

63 1 3 

636. 

6409 

6457 

65o5 

6553 

6601 

48 

6649 

6697  !  6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

906 

71 2d 

7176  ,  7224 

7272 

7320 

7368' 

74.6 

7464 

75.2 

7539 

48 

907 

7607 

7655  I  7703 

775i 

7799 

iu 

7V 

7942 

7990 
84b8 

8o38 

48 

908 
909 

80S6 

8i34  !  8181 

8229 

8277 

8373 

8421 

85.6 

48 

8564 

8612   8659 

8707 

8755 

88o3 

8850 

8898 

8946 

8994 

48 

910 

959041 

9089  91.37 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

48 

911 

9318 

9566   9614 

9661 

9709 

9757 

9^04 

9S32 

991)0 

9947 

48 

9'2 
913 
914 

9995 

••42  •   ••90 

•.38 

•.83 

•233 

•280 

•328 

•376 

•423 

48 

96047 1 

o5i8  1  o566 

o6i3 

0661 

0709 

0756 

0804 

085. 

0899 

48 

0946 

0994  1  1041 

1.36 

1.84 

.23. 

1279 

.326 

'I'^i 

47 

915 

1421 

1469   i5i6 

1 563 

.6.1 

i658 

1706 

1733 

1801 

1848 

47 

916 

1895 

1943  1  1990 

2o38 

2o85 

2.32 

2180 

2227 

2275 

2322 

47 

918 

2369 
2843 

2417  1  2464 

25ll 

2559 

2606 

2653 

2701 

2748 

2793 

47 

2890  1  2937 

2985 

3o32 

3079 

3.26 

3.74 

3221 

3268 

47 

919 

33i6 

3363  1  3410 

3457 

35o4 

3552 

3599 

3646 

3693 

3741 

47 

920 

963788 

3835  '  3882 

3929 

4448 

4024 

4071 

4.18 

4.65 

42.2 

47 

921 

4260 

4307 

4354 

4401 

4495 

4542 

4590 

4637 

4684 

47 

922 

4731 

4778 

4825 

4872 

49'9 

4966 

5oi3 

5o6i 

5io8 

5.55 

47 

923 

5202 

5240 

5296 

5343 

5390 

5437 

5484 

553. 

5578 

5625 

47 

924 

567  2  j  5719 

5766 

58i3 

5,860 

5907 

5934 

600. 

6048 

6095 

47 

925 

6142!  6180 

6236 

6283 

6329 

6376 

6423 

6470 

?'il 

6564 

47 

926 

6611 

6658  6705 

6702 

6799 

6845 

6892 

6939 

6986 

7033 

47 

927 

7080 

7127  i  7173 

7220 

7267 
8203 

73.4 

736. 

7408 

7434 

730. 

47 

928 

7548 

7595 

It^ 

7688 

7782 

7829 

7875 

7922 

7969 

47 

929 

8016 

8062 

81 56 

8249 

8296 

8343 

8390 

843b 

*47 

93o 

968483'  853o 

8576 

8623 

8670 

87,6 

8763 

88.0 

8856 

8903 

47 

93. 

89501  8996 
9416  9463 

9043 

9090 

9.36 

9.83 

9229 

9276 

9323 

9:559 

47 

932 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9S33 

47 

933 

9B82;  9928 

9070 

••21 

••68 

•ii4 

•.61 

•207 

•234 

•3oo 

^1 

934 

97o347i  0393  i  0440 

0486 

o533 

0579 

0626 

067  J 

°''i? 

0763 

46 

935 

0812  oSd8  i  0904 
1276'  i322  1  1369 

0931 

0997 

1044 

1090 

1.37 

ii83 

1229 

46 

936 

14.5 

1461 

i5o8 

1 554 

160. 

1647 

J  693 

46 

937 

1740'  1786  i  i832 

1879 

.923 

1971 

20.8 

2064 

2110 

2.37 

46 

938 

22o3  2249   2295 

2342 

2388 

2434 

248. 

2327 

2573 

2619 

46 

939 

2666  2712   2738 

2804 

285i 

2897 

2943 

2989 

3o33 

3082 

46 

N. 

0   1    I    1   2 

3 

^ 

5 

6 

7 

' 

9 

D. 

25 


16 


A   TABLE    OF   LOGARITHMS   FROM    1    TO   10,000. 


i  N. 

1 

0 

I   i   •>   1   3     4     5     6 

7     ^     9 

D. 

1 

i  940 

973128 

3174  3220  j  3266  33i3  3359 

,  34o5 

:  345i  1  3497  '  3543 

46 

i  941 

■  3590 

3636  1  3682   3728   3774  |  332o 

3866 

3913  1  3959  '  4oo5 

i  4374  !  4420  i  4466 

46 

'  942 

4o5i 

4097  !  4«43   4189  4235  '  4281 

i  4327 

'46 

;  943 

45i2 

4558  :  4604  465o   4696  4742 

4788 

!  4834  i  4880  ;  4926 

46 

944 

4972 

DOlS 

5o64  i  5iio  i  5i56   6202 

3243 

1  5294  '  5340  !  5336 

46 

;  943 

54J2 

5478 

5524  I  5570  ;  56i6   5662   5707 

1  5753   5799  ■  5343 

46 

946 

5891 

3937 

t  5983  1  6029  ;  6075  61 2 1   6167 

1  6212  1  6253  1  63o4 

46 

947 

63DO 

6396 

6442  !  648S   6533  ,  65^9  j  6625 

,  6671   6717 
j  7129  7175 

!  6763 

46 

948 

6808 

6354 

6900  6946  6992  '  7037  i  7083 
7358   74o3  .  7449  >  7493  i  7^41 

1  7220 

46 

949 

7266 

73.2 

1  7386  7632 

7678 

46 

930 

977724 

7769  '  7815  1  7861  !  7906  i  7932  7995 

8226  8272  .  83 1 7  8363   840Q  '  8434 

I  8043  i  8089 

8i35 

46 

931 

biSi 

85oo  i  8546 

859. 

46 

i  932 

8637 

S633  8728  :  8774  88 I 9 

8865   891 1 

1  8936  .   9002 

'  9047 

46 

!  953 

9093 

9i38  9184  9230  9275 

9321   9366 

;  9412  :  9437 

95o3 

46 

;  954 

9548 

9594  :  9639  :  9685  9730 

9776  9821 

•9867  9912 

9958 

46 

955 

9S0003 

0049  '  0094  '  0140  oi85 

023 1 

0276 

o322   o367 

0412 

45 

956 

oj5S 

o5o3   0349 

0594  0640 

o6S5 

0730 

0776  ;  0821 

0867 

43 

957 

0912 

0957  i  ioo3 

1048   1093 

1 139 

1184 

1229   1275 
1683  '.   1728 

l320 

43 

958 

ijbo 

141 1  1  1456 

i5oi   1 547 

1592 

1637 

1773 

45 

959 

1819 

1864  i  1909 

1954  ,  2000 

2045 

2090 

2i35  ;  2i8i 

2226 

45 

i  960 

9S2271 

23ib  !  2362 

2407  i  2432 

2497  i  2543 

2588  '  2633  I  2678 

45 

i  961 

•  2-23 

2769 

28.4 

2B59  2904 
33 ro  :  3356 

2949   2994 

3o4o  3o35  3i3o 

45 

;  962 

3175 

3220 

3265 

3401   3-146 

3491   3536  1  353 1 

45 

i  963 

3626 

3671 

3716 

3762  ,  38o7- 

3852   3.S97 

3942  ■   3987 
4392  4437 

4o32 

45 

'  964 

4077 

4122 

4167 

4212  :  4257 

43o2  ;  4347 

4482 

45 

965 

4527 

4572 

4617 

4662  '  4707 

4732  ,  4797 

4842  4887 

4932 

45 

906 

49^7 

5o22  :  5067 

5ii2  ;  5i57 

3202  1  5247 

5292   5337 

5382 

45 

i  967 

5426 

5471  ,  55i6 

556i  ;  56o6 

565 1  I  5696 

5741  '  5786 

5830 

45 

i  968 

DS75 

5920 

5965  j  6010  :  6o53 

6100  ;  6144 

6189  1  6234 

6279 

45 

969 

6324 

6369 

6413 

6458  ,  65o3 

6548  I  6593 

6637  j  6682 

6727 

43 

970 

9S6772 

6817 

6861 

6906  1  6951 

6996  j  7040 

7085  7i3o 

7175 

45 

97 « 

7219 

7264 

7309 

7353   7398 

7443  i  7488 

7532  :  7577 

7622 
806S 

45 

972 

7666 

7711 

7756 

7800  .  7843 
8247   8291 

IX  i  It. 

7979    8024 

45 

973 

8ii3 

807 

8202 

8423  :  8470 

85i4 

45 

974 
970 

8559 
9005 

8604 
9049 

8648 
9094 

8693   8737 
9i38  9i83 

8782  1  8826 
9227  1  9272 

8871  '  8916 

9316  9361 

8950 
9403 

45 
45 

976 

9430 

9494 
9939 
o3b3 

9339 

9583   9628 

9672  ;  97'7 

9761  9806 

9350 

44 

977 

9S95 

99^3 

••28   ••72 

•117  1  •161 

•206  •25o 

•294 

44 

978 

990339 

0428  ;  0472  o5i6 

o56i  !  o6o5 

o65o  0694 

0733 

44 

979 

07S3 

0827 

0871  j  0916  .  0960 

1004  i  1049 

1093  ;  1137 

1182 

44 

9S0 

991226 

1270 

i3i5  1  i359  !  i4o3 

1448  '  1492 

1 536  '  i58o 

l623 

44 

931 

1669 

1713 

1758 

1802   1846 

1890 

1935 

2377 

1979   2023 

2067 

44 

9S2 

2111 

2i56 

2200 

2244   2288 

2333 

2421   2465 

25o9 

44 

9S3 

2554 

2598  ;  2642 
3o39  ■   3o83 

2686   2730 

2774 

2819 

2863  i  2907 

2931 
3392 

44 

934 

2995 

3127   3172 

32i6 

3260 

33o4  ■   3348 

44 

985 

3436 

34S0  1  3524 

3568  36 1 3 

3657 

3701 

3745  ;  3789 

3333 

44 

9S6 

3877 

3921  1  3965 

4009  4o53 

4097 

4I4I 

4i85  :  4229 

4273 

44 

9S7 

43.7 

436 1   44o5 

4449  4493 

4537 

458i 

4625  i  4669 

47i3 

44 

9S3 

4737 

4801   4843 

4889  ,  4933 
5328  ,  5372 

4977 

502I 

5o65  5 1 08 

5i52 

44 

989 

5196 

5240  !  6284 

3416 

5460 

55o4  j  5547 

5591 

44 

990 

993635 

5679  '  5723 

5767  ■  58n 

5S54 

5898 

5942  '   5986 

6o3o 

44 

991 

6074 

6117  i  6161 

62o5  6249 

6293 

6337 

638o  6424 

6468 

44 

992 

65.2 

6555  i  6599 

6643  6687 

6731 

6774 

6818  6S62 

?S 

44 

993 

6949 

6993  j  7037 

70S0  7124 

7168 

7212 

7255  ,  7209 
7692  i  7736 
8129  :  8172 

44 

994 

73% 

7430  1  y474 

7517   7361  i  7605 

7648 

ll]l 

44 

995 

7823, 

7867  ,  79«o 
83o3  ;  8347 

7954  7993  ,  8041 

8o85 

44! 

996 

8259! 

8390  8434  j  8477 

8521 

8564  8608 

8652 

441 

997 

8695 

8739  1  8782 

8826  8869  !  8913 

8956 
9392 

9000  9043 

9087 

44  i 

998 

9.31 

9174  92«8 

9261   93o5  i  934S 

9435  :  9479 

9522 

44: 

999 

9565 

9609  :  9602 

9696  9739  i  9783 

9826 

9870  9913 

9957 

43  i 

N. 

0   ; 

I  '  2  ;  3   4    5 

6 

7  ,  8 

9  1 

D. 

A  TABLE 

OF 

LOGARITHMIC 
SINES   AND   TANGENTS 

FOR  EVERT 

DEGREE  AND  MINUTE 
OF  THE  QUADKANT. 


Remark.  The  minutes  in  tlie  left-liand  column  of  eacli 
page,  increasing  downwards,  belong  to  tlie  degrees  at  the 
top ;  and  those  increasing  upwards,  in  the  right-hand  column, 
belong  to  the  degrees  below. 


18 


(0    DEGREES.)      A   TABLE   OF    LOGARITHMIC 


,  M. 

Sine 

D. 

'    CoPine  • 

D. 

Tang. 

D. 

Cotang. 

o 

0 • 000000 

10- 000000  j 

0- 000000 

Infinite. 

60 

I 

6-463726 

5017-17 

000000  1 

•00 

6-463726 

5017-17 

i3^536274 

^ 

2 

764756 

2934-85 

000000  1 

-oo 

764756 

2934-83 

235244 

3 

940S47 

2082.31 

000000  1 

-00 

940847 

2082-31 

059153 

U\ 

4 

7-065786 

i6i5-i7 

0000  :;o  j 

-oo 

7.065786 

i6i5-i7 

12^934214 
63731)4 
758122- 

5 

162696 

i3i9-68 

oooooo 

•  00 

162696 

55  j 

6 

241877 

1115.75 

9 -999999 

•OI 

241878 

54  ; 

I 

308S24 
366816 

\^,t 

999999 
999999 

•01 

-01 

30S825 
366817 

691175 
633 183 

53  , 

52  i 

9 

417968 

1  762-63 

999999 
999998 

•01 

417970 

762-63 

582o3o 

5i  1 

lO 

463^5 

689.88 

.01 

463727 

6S9-88 

5362-3 

5o  ! 

II 

7.5o5ii8 

629-81 

9999998 

.01 

7.5o5i2o 

629.81 

12.494880 
457091 

^\ 

12 

542906 

.  5-9.36 
536-41 

999997  i 

•OI 

542909 

579.33 

i3 

577668 

999997  1 

-01 

577672 

536-42 

422328 

47 

14 

609853 

1  499-33 

999996 

-01 

609857 

Z't 

390143 

46 

i5 

6398 I 6 

467-14 

999996 

-OI 

639820 

36oi8o 

45 

i6 

667845 

438.81 

999995  j 

.01 

667849 

438-82 

332I3I 

44 

]l 

694173 

413.72 

999995 

•01 

694179 

413.73 

3o582i 

43 

718997 

391-35 

999994 

.01 

719004 

391.36 

257316 

42 

»9 

742477 

371-27 

999993 

•01 

7424«4 

371-28 

41 

20 

764754 

353-15 

999993 

•01 

764761 

351.36 

235239 

40 

21 

7-785943 

336.72 

9-999992 

•01 

7.785951 

336.73 

12.214049 

l^ 

22 

806146 

321-75 

999991 

•01 

806155 

321.76 

193843 

23 

825451 

3o8.o5 

-01 

825460 

3o8.o6 

174540 

37 

24 

843934 

itu 

999989 

-02 

843944 

295-49 
283-90 

i56o56 

36 

25 

861662 

9999S8 

.02 

8616-4 

138326 

35 

26 

'& 

273-17 
263-23 

999.988 

•02 

878708 

273-18 

121292 

34 

11 

999987 

-02 

895099 

263-25 

104901 

33 

910879 

1%:^ 

999986 

-02 

910894 
926134 

254-01 

089106 
073866 

32  1 

29 

9261 19 

9999S5 

-02 

245-40 

3i 

3o 

940842 

237.33 

999983 

.02 

940S58 

237.35 

059142 

3o 

3i 

7-955082 

220-80 

9-999982 

•02 

7.955100 

229.81 

12.044900 

It 

32 

968870 

222-73 

999981 

•02 

968889- 
982253 

222-75 

o3iiii 

33 

982233 

216-08 

999980 

.02 

216-10 

017747 

27 

34 

995198 

8-007787 

209-81 

999979 

•02 

995219 

209-83 
203-92 
198-33 

004781 

26 

35 

203-90 
198.31 

999977 
999976 

•02 

8-007809 
020045 

11.992191 
968055 

25 

36 

020021 

•02 

24 

u 

031919 
04350 I 

193-02 
188.01 

999975 

.02 

o3iq45 
043527 

193-05 
188.03 

23  1 

999973 

.02 

956473 

22  ; 

39 

054761 

183-25 

999972 

•02 

054809 

183.27 

945191 

21  '■ 

40 

065-76 

178.72 

999971 

•02 

o65So6 

178.74 

934194 

20 

41 
42 

8-0-6500 
086965 

174-41 
170.31 

9.999969 
999968 

02 
.02 

8-076531 
086997 

174.44 
170-34 

"■& 

\t 

43 

097 1  S3 

166.39 

999966 

.02 

097217 

166-42 

902783 

n 

44 

107167 

162.63 

999964 

•o3 

107202 

162.68 

|ji?] 

16 

45 

1 16926 

159-08 

099963 

•o3 

116963 

i59^io 
i53^68 

i5 

46 

126471 

155-66 

999961 

•o3 

i265io 

873490 

14 

% 

i358io 

152.38 

999959 
999938 

.03 

i3585i 

132-41 

864149 

i3 

144953 

149-24 

-o3 

144C96 

149-27 

146.27 

143.36 

855004 

12 

49 

I 6268 I 

146-22 

999  ,,53 

•o3 

I 53932 

846048 

11 

56 

143-33 

999954 

•o3 

162727 

83-273 

10 

5i 

8-171280 

140.54 

9-999952 

•o3 

8.171328 

140-57 

11.828672 

I 

52 

1-9713 

137-86 

999950 

•o3 

179763 
i88o36 

\lir. 

820237 

53 

187985 

135.29 

990948  1 

•o3 

811964 
803844 

I 

54 

I01O2 

i32.8o 

999946  1 

.o3 

196156 

132-84 

55 

204070 

i3o.4i 

999944  ! 

•o3 

204126 

i3o.44 

« 

5 

56 

^11.895 

128-10 

999942  1 

•04 

211953 

128-14 

4 

u 

219581 

125-87 

999940  1 

•04 

219641 

125-90 

780359 

3 

227134 

123.72 

999938  1 

.04 

227195 

123-76 

772803 

2 

59 

234557 

121-64 

999936  ; 

.04 

234621 

121-68 

765379 

I 

60 

241855 

119-63 

999934 

•04 

241921 

119-67 

758079 

0 

Cosine 

J). 

Sine 

Cotang. 

D. 

Tang. 

M. 

DEGREES.) 


SINES   AND   TANGENTS.       (1    DEGREE.) 


19 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D 

Cotang. 

0 

8-24i855 

119-63 

9.999934 

04 

8.241921 

1x9 

67 

11.758079  60 

I 

249033 

117 

68 

999932 

04 

249102 

117 

72 

750S98   5o 
743835   58  ' 

2 

256094 

ii5 

80 

999929 

04 

256i65 

1x5 

84 

3 

263042 

ii3 

98 

999927 

04 

263ii5 

114 

02 

736885   57 

4 

2698S1 

112 

21 

99992 D 

04 

269906 

112 

25 

730044   56 

5 

276614 

no 

5o 

999922 

04 

276091 

no 

54 

723309  1  55 

6 

283243 

108 

83 

999920 

04 

283323 

108 

87 

716677  1  54 

7 

289773 

107 

21 

999918 

04 

289806 

107 

26 

7x0144  i  53 

8 

296207 

io5 

65 

9999 r 5 

04 

296292 

io5 

70 

703708  i  52 

9 

302546 

104 

i3 

9999 '3 

04 

302634 

104 

x8 

697366 

5i 

10 

308794 

102 

66 

999910 

04 

308884 

102 

70 

691x16 

5o 

II 

8-3i49o4 

lOI 

22 

9.999907 

04 

8.3i5o46 

lOX 

26 

11.684954 

% 

12 

321027 

99 

82 

99990D 

04 

321X22 

99 

87 

678878 

i3 

327016 

98 

47 

999902 

04 

327XX4 

98 

5i 

672886 

47 

14 

332924 

97 

U 

999899 

o5 

333025 

97 

'9 

666975 

46 

i5 

338753 

95 

86 

999^97 

o5 

338856 

95 

90 

661x44 

45 

i6 

344304 

94 

60 

999894 

o5 

344610 

94 

65 

655390 

44 

\l 

35oi8i 

93 

38 

999891 

o5 

3502M9 
355895 

93 

43 

649711 

43 

355783 

92 

2 

999888 

o5 

92 

24 

644105 

42 

^9 

36i3i5 

% 

999885 

o5 

36x43o 

% 

08 

638570 

41 

20 

366777 

90 

999882 

o5 

366895 

95 

633io5 

40 

21 

8-372I7I 

88 

80 

9.999879 

o5 

8. 372292 

88 

85 

11.627708 

39 

22 

377499 

87 

72 

999876 

o5 

377622 

87 

77 

622378 

38 

23 

382762 

86 

67 

999873 

o5 

382889 

86 

72 

6171x1 

37 

24 

387962 

85 

64 

999870 

o5 

388092 

85 

70 

61x908 

36 

25 

393101 

84 

64 

999867 

o5 

393234 

84 

70 

606766 

35 

26 

398179 

83 

66 

999864 

o5 

3983x5 

83 

71 

60x685 

34 

27 
28 

403x99 

82 

71 

999861 

o5 

403338 

82 

76 

596662 

33 

408161 

81 

77 

999858 

o5 

4o83o4 

8x 

82 

591696 

32 

29 

4i3o68 

80 

86 

999854 

o5 

4i32i3 

80 

91 

586787 

3i 

3o 

417919 

79 

96 

?9985i 

06 

4x8068 

80 

02 

581932 

3o 

3i 

8.422717 

]§ 

2? 

9.999848 

06 

8.422869 
427618 

]l 

14 

11.577131 

^1 

32 

427462 

999844 

06 

3o 

572382 

33 

432156 

11 

40 

999841 

06 

4323x5 

77 

45 

567685 

'2 

34 

436800 

76 

57 

999838 

06 

436962 

76 

63 

563o38 

35 

441394 

75 

77 

999834 

06 

441 56o 

75 

83 

558440 

25 

36 

445941 

74 

99 

999831 

06 

446x10 

75 

o5 

553890 

24 

ll 

450440 

74 

22 

999827 

06 

45o6i3 

74 

28 

549387 

23 

454893 

73 

46 

999823 

06 

455070 

73 

52 

544930 

22 

39 

459301 

72 

73 

999820 

06 

45948 X 
463849 

72 

79 

540019 

21 

40 

463665 

72 

00 

999816 

06 

72 

06 

536x5i 

20 

41 

8.467985 

71 

29 

9.999812 

06 

8.468172 

71 

35 

11.53x828 

10 
18 

42 

472263 

ll 

60 

999809 
999805 

06 

472454 

70 

66 

527546 

43 

ttnt 

91 

06 

476693 

69 

98 

523307 

;? 

44 

ll 

24 

999801 

06 

480892 
485oDo 

6n 
68 

3x 

519x08. 

45 

484848 

59 

999797 

07 

65 

5x4900 

i5 

46 

488963 

67 

t 

999793 

07 

489170 
493230 

68 

01 

5io83o 

x4 

47 

493040 

67 

999790 
999786 

07 

67 

38 

506700 

i3 

48 

497078 

66 

% 

07 

497293 

66 

76 

502707 

X2 

49 

5oio8o 

66 

999782 

07 

50x298 

66 

x5 

498702 

11 

5o 

5o5o45 

65 

48 

999778 

07 

505267 

65 

55 

494733 

10 

5i 

'tX", 

64 

89 

9-999774 

07 

8.509200 

b'A 

96 

11-490800 
486Q02 

I 

52 

64 

3i 

999769 
999765 

07 

513098 

64 

39 

53 

516726 

63 

75 

07 

516961 

63 

82 

483o39 

1 

54 

52o55i 

63 

19 

999761 

07 

520790 

63 

26 

479210 

6 

55 

524343 

62 

64 

999757 

07 

524586 

62 

72 

470414 

5 

56 

528102 

62 

II 

999753 

07 

528349 

62 

18 

47i65i 

4 

u 

531828 

61 

58 

999748 

07 

532080 

6x 

65 

467920 

3 

535523 

61 

06 

999744 

07 

535779 

61 

i3 

464221 

2 

59 

539186 

60 

55 

999740 

07 

54^084 

60 

62 

460553 

I 

60 

542819 

60 .  04 

999735 

07 

60- 1.^ 

456916 

0 

Cosine 

D. 

Sine 

Cotang. 

D. 

Tang 

If) 


(88    DEGREES.) 


20 


(2    DEGREES.)       A   TABLE   OF    LOGARITHiMIO 


M. 

Sine 

D. 

Cosine  t 

D. 

Tang. 

D. 

Cotang.  1 

8.542819 

60 

•  04 

9-9Q9735  ' 

•07 

8.543084 

60.12 

11-436916  i  60 

546422 

59 

55 

999731 

07 

546691 

59.62 

453309  1  5q 

549995 

59 

06 

999726 

:S 

550268 

59.  14 

58.66 

449732  ;  58 

553539 

5« 

58 

999722 

553817 

446183  \   57 
442664  I  56 

557054 

53 

II 

999717  I 

.08 

557336 

58-19 
57-73 

5 

560340 

57 

65 

999713  1 

08 

560S28 

439172  1  55 

6 

563999 

57 

'9 

999708 

08 

564291 

57-27 

433709  1  54 
432273  1  53 

I 

567431 

56 

74 

999704  1 

08 

567727 

56-82 

570836 

56 

3o 

999699  i 

08 

571137 

56-38 

428863  }  52  1 

9 

574214 

55 

S7 

999694  1 

08 

574520 

lit 

425480  i  5i  I 

10 

577566 

55 

44 

9996S9  1 

08 

577877 

422123   5o 

II 

8.580S92 

55 

02 

9.999685  1 

08 

8.581208 

55-10 

11-418702  49 
4154S6  48 

12 

584193 

54 

60 

^^80 

08 

5843x4 

54-68 

i3 

587469 

54 

19 

999675 

08 

587795 

54-27 

412203   47 

14 

590-^21 

53 

"9 

999670 

08 

091031 

53-87 

408949  46 

i5 

593948 

53 

39 

999663 

08 

594283 

53-47 
53-08 

405717  1  45 

i6 

597132 

53 

00 

999660 

08 

597492 

402 5o8 

44 

\l 

6oo332 

52 

61 

999655  1 

08 

600677 

52-70 

399323 

43 

6034^9 

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23 

999630  j 

08 

603839 
606978 

52-32 

396I6I 

42 

19 

606623 

5i 

86 

999645 

09 

5i-94 
51-58 

398022 

41 

20 

609734 

5i 

49 

999640 

09 

610094 

40 

21 

8.612S23 

5i 

12 

9.999635 

09 

8.613189 

5l.2I 

II-3868II 

39 

22 

613891 

5o 

76 

999629 

09 

616262 

5o-85 

383738 

38 

23 

61S937 

5o 

41 

999624 

09 

6i93i3 

5o.5o 

380687 

37 

24 

621962 

5o 

06 

999619 

09 

622343 

5o-i5 

377657 
374648 

36 

25 

624963 

49 

72 

999614 

09 

625352 

49-81 

35 

26 

627948 

49 

38 

999608 

09 

628340 

49-47 

371660 

34 

11 

63091 1 
633854 

49 

04 

999603 

09 

63i3o8 

4q-i3 
48-80 

368692 

33 

48 

71 

999397 

09 

634236 

365744 

32 

29 

636776 

48 

39 

999592 

09 

637184 

48-48 

3628:6 

3i 

So 

639680 

48 

06 

9995b6 

09 

640093 

48-16 

359907 

3o 

3i 

8-642563 

47 

75 

9.999581 

09 

8.642982 

47-84 

11-357018 

20 

32 

645428 

47 

43 

999575 

09 

645853 

47-53 

354147  1  28 

33 

648274 

47 

12 

999370 

09 

648704 

47-22 
46-91 

331296  '  27 

34 

63II02 

46 

82 

999564 

09 

65i537 

348463  !  26 

35 

65391 1 

46 

52 

^558 

10 

654352 

46-61 

345648   25 

36 

636702 

46 

22 

999553 

10 

657149 
65992§ 

46-3i 

34285 I   24 

37 

639475 

43 

92 

999547 

10 

46-02 

340072   23 

38 

662230 

43 

63 

999541 

10 

662689 

45-73 

337311   22 

39 

664968 

43 

35 

^535 

10 

665433 

43-44 

334367  1  21  i 

40 

6676S9 

45 

06 

999329 

10 

668160 

45-26 

331840 

20 

41 

8.670393 

44 

79 

9.999524 

10 

8.670870 

44-88 

n  32gi3o 

\t 

42 

6730S0 

44 

5i 

999518 

10 

673563 

44-61 

326437 

43 

673751 

44 

24 

999312 

10 

676239 

44-34 

323761 

17  i 

44 

678405 

43 

97 

999306 

10 

678900 

44-17 

321 100  i  16 

45 

681043 

43 

70 

999500 

10 

681344 

43-80 

3i8456  I  i5 

46 

683665 

43 

44 

999493 

10 

684172 

43-54 

3i5828i  14 

47 

686272 

43 

18 

999487 

10 

686784 

43-28 

3i32i6   i3 

48 

688863 

42 

92 

999481 

10 

689381 

43-03 

310619  1  12  i 

49 

691438 

42 

67 

999475 

10 

691963 

42-77 

3o8o37 

II 

5o 

693998 

42 

42 

999469 

10 

694329 

42-52 

3o547i 

10 

5i 

8.696543 

42 

17 

9.999463 

II 

8.697081 

42.28 

"13 

I 

32 

699073 

41 

92 

999456 

II 

699617 

42.  o3 

53 

701589 

41 

68 

999450 

II 

702139 

41-79 

297861 

7 

54 

704090 

41 

44 

999443 

II 

704646 

41-53 

295354 

6 

55 

706377 

41 

21 

999437 

II 

707140 

41-32 

292860 

5 

56 

709049 

40 

97 

999431 

II 

709618 

41 -08 

200382 

287917 

4  ! 

57 

71 1307 

40 

74 

999424 

II 

7120S3 

40-85 

3  1 

58 

713932 

40 

5i 

999418 

II 

714534 

40-62 

285465 

2 

09 

716383 

40 

29 

99941 1 

II 

^:^^6 

40-40 

283028 

I 

60 

718800 

40-06 

999404 

" 

40-17 

280604 

0 

Cosine 

D. 

Sine 

Cotang. 

D. 

Ting. 

M. 

(87  DEGRKES.) 


SINES   AND   TANGENTS.       (3   DEGREES.) 


21 


M. 

Sine 

D. 

Co  ine 

D. 

Tang. 

D. 

Cotana:.  I 

0 

8-718800 

40.06 

9-999404 

II 

8-719396 

40-. 7 

1 1 . 280604 

60 

1 

721204 

39 

84 

999398 

II 

721806 

39-95 

278.94 

U 

2 

723595 

39 

62 

999391 

II 

724204 

39-74 

273796 

3 

725972 

39 

41 

999384 

II 

726588 

39-52 

273412  i  57 

4 

728337 

39 

«9 

999378 

II 

72«959 
73i3i7 

39.30 

27.041  1  56 

5 

730688 

38 

98 

■  99937' 

II 

38.68 

268683  1  5i) 

6 

733027 

38 

77 

999364 

12 

733663 

266337  1  54 

I 

735354 

38 

■'^7 

9993)7 

12 

735996 

264004  1  53 

737667 

38 

36 

999350 

12 

7383.7 

38.48 

261683   52 

9 

739969 

38 

16 

999313 

12 

740626 

38.27 

259374  5 1 

10 

742259 

37 

96 

999336 

12 

742922 

38.07 

257078   5o 

II 

8-741536 

37 

76 

9-999329 

12 

8-745207 

37.87 

11.254793   4Q 

252321  1  48 

12 

746802 

37 

56 

999322 

12 

747479 

37.68 

i3 

749055 

37 

37 

9993 1 5 

12 

749740 

37-49 

25o26o 

47 

14 

75(297 

37 

17 

999308 

12 

75.989 

37-29 

24801  I 

46 

i5 

753528 

36 

98 

999301 

12 

734227 

37-10 

245773 

45 

i6 

755747 

36 

79 

999294 

12 

756453 

36.92 

243547 

44 

\l 

757955 

36 

61 

999286 

12 

758668 

36.73 

241332 

43 

7601 5 1 

36 

42 

999279 

12 

760872 

36-55 

239.28  ;  42 

19 

762337 

36 

24 

999212 

12 

763o65 

36.36 

236935  1  41 

20 

7645 II 

36 

06 

999265 

12 

765246 

36.18 

234754 

40 

21 

8.766675 

35 

83 

9-999257 

12 

8-767417 

36-00 

11-232583 

? 

^^ 

768828 

35 

70 

999250 

i3 

769578 

35-83 

230422 

23 

770970 

35 

53 

999242 

i3 

771727 

35-65 

228273 

37 

24 

773101 

35 

35 

999235 

i3 

773866 

35-48 

226.34 

36 

25 

775223 

35 

18 

999227 

i3 

775995 

35-3i 

224oo5 

35 

26 

777333 

35 

01 

999220 

i3 

778114 

35-14 

221886 

34 

27 

779134 

34 

84 

999212 

i3 

780222 

34-97 

2.9778 

33 

28 

781524 

34 

67 

999205 

i3 

782320 

34 -80 

217680 

32 

^9 

7^-^3605 

34 

5i 

999197 

i3 

784408 

34.64 

2.5392 

3i 

So 

785675 

34 

3i 

9991 89 

i3 

786486 

34-47 

2.35i4 

So 

3i 

8.787736 

34 

18 

9.990181 

i3 

8-788554 

34-3. 

II. 2 1 1446 

11 

32 

789787 

34 

02 

999174 

i3 

7906.3 

34-.5 

209387 

33 

795^28 

33 

86 

999166 

i3 

792662 

33-09 
33-83 

207338 

27 

34 

793859 

33 

70 

999158 

i3 

794701 

205299 

26 

35 

795881 

33 

54 

999 1 5o 

i3 

79678. 

33-68 

203269 
201248 

25 

36 

797894 

33 

39 

999 1 42 

i3 

798752 

33-52 

24 

ll 

799^97 
801S92 

33 

23 

999 '34 

i3 

800763 

33-37 

199237 

23 

38 

33 

08 

999126 

1 3 

802765 

33-22 

197235  1  22 

39 

803876 

32 

93 

Q99118 

i3 

804758 

33-07 

195242  1  21 

40 

8o5852 

32 

78 

9991 10 

i3 

806742 

32-92 

193238 

20 

41 

8.80^819 

32 

63 

9-999102 

i3 

8-808717 

32-78 

II •191283 
189317 

;i 

42 

809777 

32 

49 

999094 

14 

8.0683 

32-62 

43 

811726 

32 

34 

999086 

14 

8.2611 

32-48 

187359  1  17  1 

44 

8.3667 

32 

i? 

999077 

14 

8.4589 

32-33 

185411 

16 

45 

815599 

32 

999069 

14 

8.6529 

32-19 

1 8347 1 

15 

46 

817522 

3i 

91 

999061 

14 

8.8461 

32-03 

18.539 

T4 

H 

8.9^^36 

3i 

77 

999053 

14 

820384 

31-91 

1796.6  '  .3 

48 

821343 

3i 

63 

999Q-''4 

14 

822298 

31-77 

177702  '  .2 

^9 

823240 

3i 

49 

999036 

14 

824205 

3i-63 

175795  1  u 

5o 

825 1 3o 

3i 

35 

999027 

14 

826.03 

3i-5o 

173S97  1  10 

5i 

8.827011 

3i 

22 

9.999019 

14 

8-827992 

3. -06 

1 1. 1 72008 

I 

52 

828884 

3i 

08 

999010 

14 

829874 

3.-23 

170.26 

53 

830749 

3o 

95 

999002 

14 

83.748 

3i-io 

.68252 

I 

54 

832607 

3o 

82 

998993 

14 

8336.3 

30-96 
30-83 

166387 

55 

834456 

3o 

69 

998984 

14 

835471 

164529 

5 

56 

836297 

3o 

56 

998976 

14 

837321 

30-70 

i626-?9 

4 

il 

838 1 3o 

3o 

43 

998967 

i5 

839163 

30.57 

160837 

3 

58 

839956 

3o 

3o 

998958 

i5 

840998 
842825 

30.45 

139002 

2 

59 

841774 

3o 

17 

99^950 

i5 

30.32 

1 57 1  75 

I 

60 

843585 

30-00 

998911 

i5 

844644 

30.19 

155336 

0 

Cosine 

D. 

Sine 

Cotang.  1 

D.   1 

Tang. 

M. 

(86    DEGREES.) 


22 


(4    DEGREES.)      A   TABLE    OF    LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang. 

0 

8-843585 

3o-o5 

9-998941 

•  15 

8-844644 

3o.i9 

11.155356 

60 

, 

845387 

•  29-Q2 
29-80 

99^932 

•  15 

846455 

30.07 

153545 

5§ 

2 

847183 

99S923 

•i5 

84H260 

29.05 

i5i74o 

3 

848971 

29-67 

99«9'4 

•  15 

85oo57 

29.82 

149943 
148154 

^I 

4 

85o75i 

29-55 

998005 
998896 

•  15 

85 1 846 

29-70 

56 

5 

852525 

29-43 

-15 

853628. 

29-58 

146372 

55 

6 

854291 

29-31 

99^887 

•  i5 

855403 

29-46 

144597 

54 

I 

856049 

20-19 

998878 

-15 

857I7I 

29-35 

142S20 
141068 

53 

857801 

29-07 
28.96 
28-84 

998S69 

•  i5 

858932 

29-23 

52 

9 

859546 

998860 

-15 

8606S6 

29-11 

139314 

5i 

10 

861283 

998851 

•  15 

862433 

29-00 

137567 

5o 

II 

8-863014 

28-73 

9-99«84i 

-i5 

8.864173 

28-88 

II. 135827 

tt 

12 

864738 

28-61 

998832 

.i5 

865906 

28.77 

134094 

i3 

866455 

28-50 

998823 

.16 

867632 

28.66 

132368 

47 

14 

8681 65 

28-39 
28-2S 

998813 

.16 

869351 

28.54 

I 30649 

46 

i5 

86986S 

99S804 

.16 

871064 

28.43 

128936 

45 

i6 

871565 

28-17 

998795 
998785 

.16 

872770 

28.32 

127230 

44 

\l 

873255 

28-06 

.16 

874469 

28-21 

125531 

43 

874938 

l]t 

998776 

.16 

876162 

28.11 

123838 

42 

19 

876615 

998766 

.16 

877849 

28.00 

I22l5l 

41 

20 

878285 

27-73 

998757 

•  16 

879529 

27.89 

1 2047 1 

40 

21 

8-879949 

27-63 

9-998747 

.16 

8-881202 

27.68 

1 1. 1 18798 

89 

22 

881607 

27-52 

998738 

•  16 

882869 

Ii7i3i 

38 

23 

883258 

27-42 

998728 

•  16 

884530 

27-58 

1x5470 

37 

24 

884903 
886542 

27-31 

998718 

.16 

8861 85 

27-47 

Ii38i5 

36 

25 

27-21 

998708 

•16 

887833 

27.37 

112167 

35 

26 

888174 

27-11 

998699 

•16 

889476 

27-27 

iio524 

34 

11 

889801 

27-00 

998689 

•16 

891112 

27-17 

108888 

33 

891421 

26-90 
26-80 

998679 

•16 

892742 

27.07 

107258 

32 

29 

893035 

998669 

•17 

894366 

Itl] 

I05634 

3i 

3o 

894643 

26-70 

998659 

•17 

895984 

104016 

3o 

3i 

8-896246 

26-60 

9-998649 

•n 

8.897596 

26.77 

1 1. 102404 

ll 

32 

8Q7842 

26-51 

998639 

•n 

899203 

26.67 

100797 

33 

899432 

26-41 

998629 

•17 

900803 

26.58 

099197 

27 

34 

901017 

26-31 

998619 

•n 

902308 
903987 
905570 

26-48 

097602 

26 

35 

902596 

26-22 

998609 

•17 

26-38 

096013 

25 

36 

904169 

26-12 

998599 

•17 

26.29 

094430 

24 

ll 

905736 

26 -03 

99858Q 
998578 

•17 

907147 

26.20 

092853 

23 

907297 

25-93 
25-84 

•>7 

90S719 

26.10 

091281 
088154 

22 

39 

908853 

998568 

•n 

9102S5 

26.01 

21 

40 

910404 

25.75 

998558 

•17 

91 1846 

25.92 

20 

41 

8.91 1949 
913488 

25-66 

9-998548 

•  17 

8-913401 

25.83 

11-086599 

;? 

42 

25-56 

998537 

•  17 

914951 

25.74 

o85o49 
o835o5 

43 

9l5022* 

25-47 

998527 

•  17 

916495 

25-65 

17 

44 

9i655o 

25-38 

998516 

•  18 

918034 

25-56 

081966 

16 

45 

918073 

25-29 

998506 

•  18 

919568 

25.47 

080432 

i5 

46 

919591 

25-20 

^^1 

.18 

921096 

25.38 

078904 
077381 

14 

47 

921 io3 

25-  12 

•  18 

922619 

25. 3o 

i3 

x48 

922610 

25-03 

998474 

•  18 

924136 

25.21 

075864 

12 

49 

9241 12 

24-94 

998464 

.18 

925649 

25.12 

074351 

II 

5o 

925609 

24-86 

998453 

•  18 

927156 

25 -o3 

072844 

10 

5i 

8-927100 

24-77 

9-998442 

.18 

8-928658 

24-q5 
24-86 

II. 071342 

t 

52 

928587 

24-69 

998431 

.18 

930155 

069845 
068353 

53 

930068 

24-60 

998421 

.18 

931647 

24-78 

7 

54 

93 1 544 

24-52 

998410 

•  18 

933134 

24-70 

066866 

6 

55 

9330 1 5 

24-43 

998399 

.18 

934616 

24-61 

065384 

5 

56 

934481 

24-35 

998388 

.18 

936093 

24-53 

063907 

4 

57 

935942 

24-27 

998377 

.18 

937565 

24-45 

062435 

3 

58 

937398 

24-19 

098366 

.18 

939032  i 

24-37 

060968 
059306 
058048 

2 

59 

938850 

24-11 

^8355  i 

•  18 

940494  1 

24-3o 

I 

60 

940296 

24 -03 

998344  1 

.18 

941952  I 

24-21 

0 

Cosine 

D. 

Sine   1 

1 

Cotang.  i 

D. 

Tang,   i  M.  j 

(85  DEGREES.) 


SINES   AND   TANGENTS.       (5    DEGREE.) 


23 


M. 

T  Sine 

D. 

Cosine 

D. 

Tano^. 

D. 

Cotang. 

o 

8-940296 
1   941738 

24-o3 

9- 99^^344 

.19 

8-941932 

24^21 

1 1 • 038048  60 

I 

23-94 
23-87 

998333 

•  19 

943404 

24-i3 

056596  5o 
o55i48   58 

2 

1   943174 

998322 

.19 

944852 

24 -o5 

3 

944606 

23-79 

998311 

•19 

946295 

23-97 

053705   57 

4 

946034 

23-71 

998300 

•19 

947734 

23-90 

052266   56 

5 

947406 

23-63 

9982^9 

•  19 

949168 

23-82 

o5o832   55 

6 

948874 

23-55 

998277 

.19 

950097 

23-74 

049403   54 

7 

!      900287 

23-48 

998266  1 

.19 

952021 

23-66 

047979   53 

8 

'      951696 

23-40 

998255 

.19 

953441 

23-60 

046559   52 

9 

i      953100 

23-32 

998243 

.19 

954S06 

23-5i 

o45i44  5i 

lo 

954499 

23-25 

998232 

.19 

906267 

23-44 

043733 

5o 

ii 

:  8-955«94 

2-3-17 

9-998220 

•19 

8-957674 

23-37 

ii^o42326 

.% 

12 

957284 

23-10 

998209  1 

.19 

909073 

23-29 

040925 

i3 

938670 

23-02 

998107  j 

998186  ! 

.19 

960473 

23-23 

039327 

% 

M 

960002 

22-95 

22-88 

.19 

961866 

23-14 

o38i34 

15 

1   961429 

998174  i 

.19 

963200 

23-07 

036745 

43 

i6 

1   962801 

22-80 

998163  ' 

.19 

964639 

23-00 

o3536i 

44 

\l 

!   964170 

22-73 

998101  i 

.19 

966019 

22-93 

22-86 

033981 

43 

1   965534 

22-66 

998139 

.20 

967394 
968766 

032606 

42 

'9 

1   966893 

22-59 

998128  , 

•  20 

22-79 

o3i234 

41 

2o 

!   968249 

22-02 

998116  ; 

.20 

970133 

22-71 

029867 

40 

21 

8-969600 

22-44 

9.998104  i 

•  20 

8-971496 

22-65 

n^o285o4 

39 

22 

970947 

22-38 

9980^2 

•  20 

972833 

22-57 

027145 

38 

23 

972289 

22-31 

•  20 

974209 

22-5l 

025791 

37 

24 

973628 

22-24 

998068  ' 

.20 

973360 

22-44 

024440 

36 

25 

974962 

22-17 

998006 

•20 

976906 

22.37 

023094 

35 

26 

976293 

22-10 

99«o44 

•20 

978248 

22-3o 

021752 

34 

27 

977619 

22-03 

998032 

.20 

979586 

22-23 

020414 

33 

28 

978941 

21-97 

998020 

-20 

980921 

22-17 

019079 

i^ 

?9 

9S0259 

21-90 

998008 

•20 

982251 

22-10 

017749 

3i 

3o 

981573 

21-83 

997996 

•20 

983577 

22-04 

016423 

3o 

3i 

8-982883 

21.77 

9.997980 

•20 

8-984899 

21-97 

ii^oiSioi 

^? 

32 

984189 

21-70 
21-63 

997972 

•20 

986217 

21-91 

013783 

33 

9S5491 

997959 

•20 

987032 

21-84 

012468 

^^ 

34 

9S67S9 

21-57 

997947 

•20 

988842 

21-78 

oiii58 

35 

988083 

21 -5o 

997935 

•21 

990149 

21.71 

009851 

25 

36 

989374 

21-44 

997922 

•21 

991451 

21.65 

008549 

H 

37 

990660 

21-38 

997910 

•21 

992730 

21.58 

007350 

23 

38 

991943 

2I-3I 

997897 

•21 

994045 

21.32 

005955 

22 

39 

993222 

21-25 

997885 

•21 

995337 

21.46 

0O4663 

21 

4o 

994497 

21-19 

997872 

.21 

996624 

21-40 

003376 

20 

4i 

8-995768 

2I-I2 

9.997860  , 

•21 

8-997908 

21.34 

11-002092 

19 

42 

997036 

21-06 

997847 

•21 

999188 

21.27 

000812 

18 

43 

998299 

21-00 

997835 

•21 

9^000465 

21-21 

10-999535 

n 

44 

999560 

20-94 

997822 

•21 

001738 

21-15 

99S262 

16 

45 

9 • 0008 1 6 

20-87 

997809 

•21 

oo3oo7 

21.09 

996993 

i5 

46 

002069 

20-82 

997797 

•21 

004272 

2i.o3 

995728 

14 

47 

oo33i8 

20-76 

9977^4 

-21 

005534 

20-97 

994466 

i3 

48 

004563 

20-70 

997771 

-21 

006792 

20-91 
2o^85 

993208 

12 

^9 

oo58o5 

20-64 

997708 

-21 

008047 

991953 

II 

5o 

007044 

20-58 

997745  ■ 

•21 

009298 

20^80 

990702 

10  1 

5i 

9-008278 

20-52  \ 

9.997732 

•21 

9^010546 

20^74 

10-989454 

I 

52 

009510^ 

20-46  ; 

997719 

•21  ' 

011790 
oi3o3i 

20  •68 

988210 

53 

-  010737' 

20-40 

997706 

•21 

20-62 

986969 

7 

54 

01 1962 

20-34 

997693 

•22 

014268 

20-56 

983732 

^ 

55 

oi3i82 

20-2q 

997680 

•22 

oi55o2 

2o-5i 

984498 

5 

56 

014400 

20-23  ! 

997667 

-22  I 

016732 

20^45 

983268 

4 

u 

oi56i3 

20-17   i 

997654 

-22  1 

017959 

20-40 

982041 

3 

016824 

20-12   ; 

997641 

-22  , 

019183  ' 

20-33 

980817 

2 

59 

oi8o3i 

20-06 

997628  , 

•22  ' 

02o4o3  1 

20^28 

979597 
978380 

I 

60 

019235 

20-00   ; 

997614 

•22 

021620  ! 

i 

20-23 

0 

CoBiue 

D. 

Sine 

Cotansr.  i 

D. 

Tiuiff.  ! 

M. 

(84  DEGREES.) 


2-i 


(G    DEGREES.)       A   TABLE   OF    LOGARITHMIC 


M. 

Sine 

D. 

Cosine  1 

D. 

Tang. 

D. 

Cotang. 

"H 

o 

9-019235 

20-00 

9-997614 

22 

9.021620 

20.23 

10-978380 

60 

I 

020435 

»9 

95 

997601 

22 

022^34 

20.17 

977166  !  5o 
975936  ;  58 

2 

021632 

19 

89 

997388 

22 

024044 

20. 11 

3 

022825 

>9 

84 

99-574 

22 

023231 

20.06 

974749  i  57 
973543  '  56 

4 

024016 

'9 

78 

99-561 

22 

026455 

20.00 

5 

025203 

'9 

73 

997547 

22 

027655 

19.95 

972345  :  55 

6 

0263S6 

'9 

67 

997534 

23 

02bS52 

19.90 

971 148  '  54 

I 

027567 
028744 

'9 

62 

997520 

23 

o3oo46 

19.85 

969934  '  53 

19 

57 

997307 

23 

o3i237 

19.79 

968763   52 

9 

029918 

'9 

5i 

997493 
997480 

23 

032425 

19.74 

967575  5i 

lo 

0310S9 

19 

47 

23 

o336o9 

19. 69 

966391  ;  5o 

II 

9-o32257 

.9 

41 

9-997466 

23 

9.034791 

19. 64 

10-965209  i  49 
964031  i  40 

12 

o3342i 

19 

36 

997432 

23 

0339/39 

19.58 

i3 

o34582 

'9 

3o 

997439 

23 

037144 

19.53 

962^36  47 
961684  ,  46 

U 

035741 

19 

25 

997425  1 

23 

o383i6 

19-48 

i5 

036896 

19 

20 

997411  1 

23 

039485 

19.43 

96o5i5  1  45 

i6 

o33o48 

^9 

i5 

99-397  . 

•.?3 

0406 5 I 

19-38 

959349  '   44 

n 

039197 

19 

10 

99.3^3 

23 

041813 

19.33 

958187  j  43 

i8 

040342 

:i 

o5 

9Q736g 

23 

042973 

19-28 

95-027  j  42 

19 

041485 

99 

997335 

23 

"044130 

19-23 

955&70  41 

20 

042625 

18 

94 

997341 

23 

045284 

19-18 

954716  i  40 

21 

9-043762 

18 

89 

9.997327 

24 

9.046434 

19-13 

10 -953566  I  39 

22 

044895 

18 

84 

9973 1 3 

24 

047582 

19.08 

932418  '  38 

23 

046026 

18 

79 

697299 

24 

049869 

19-03 

931273  ;  37 

24 

047154 

18 

73 

99.285 

24 

18-98 

95oi3i  1  36 

25 

048279 

18 

70 

997271 

24 

o5ioo8 

18-93 
18-89 

948992  ;  35 

947836  ;  34 

26 

049400 

18 

65 

997257 

24 

o52i44 

11 

o5o5i9 

18 

60 

997242 

24 

053277 

18-84 

946723  i  33 

o5i635 

18 

55 

997228 

24 

054407 

18.79 

945593  I  32 

29 

052749 

18 

5o 

997214 

24 

055535 

18.74 

944465  i  3i 

3o 

053859 

18 

45 

997199 

24 

056659 

18.70 

943341  j  3o 

3i 

9-054966 

18 

41 

9-997185 

24 

9.057781 

i8.65 

10.942219  1  29 
941100  1  28 

32 

o56o7i 

18 

36 

997170 

24 

058900 

18.69 

33 

057172 

18 

3i 

997136 

24 

060016 

18.65 

9399S4  ;  27 

34 

058271 

18 

27 

997'4i 

24 

o6ii3o 

18. 5i 

938870 

26 

35 

059367 

18 

22 

997127 

24 

062240 

18.46 

937760 

25 

36 

060460 

18 

17 

997112 

24 

063348 

18.42 

936652 

24 

37 

061 55 I 

18 

i3 

997098 

24 

064453 

18.37 

935547 

23 

38 

062639 

18 

08 

997083 

25 

065556 

18.33 

934444 

22 

39 

063-24 

18 

04 

99706S 

25 

066655 

18.28 

933345 

21 

40 

064806 

17 

99 

997053 

25 

067752 

18.24 

932248 

20 

41 

9-065885 

17 

94 

9.997039 

25 

9.068846 

18.19 

10-931154 

\l 

42 

066962 

17 

90 

997024 

25 

069938 

18. ID 

930062 

43 

o68o36 

17 

86 

997009 

25 

071027 

18.10 

9289-3 

]l 

44 

069107 

n 

81 

996994 

25 

072113 

18.06 

927887 

45 

070176 

n 

77 

996979 

25 

073197 

18.02 

926803 

i5 

46 

071242 

17 

"^ 

996964 

23 

074278 

17.97 

925722 

14 

% 

072306 

17 

68 

996949 

25 

075356 

:?| 

924644 

i3 

073366 

17 

63 

996934 

25 

076432 

923568 

12 

49 

074424 

17 

99 

996919 

25 

078576 

17-84 

922495 

II 

5o 

075480 

n 

5o 

996904 

25 

17-80 

921424 

ID 

5i 

9.076533 

17 

5o 

9.996889 

25 

9.079644 

17-76 

10-920356 

I 

52 

077583 

17 

46 

996874 

25 

080710 

17.72 

919290 

53 

078631 

n 

42 

996858 

25 

081773 

17-67 

918227 

I 

5 

54 
55 

079676 
080719 

17 

n 

38 
33 

996843 
996828 

25 

23 

082833 
083891 

17-63 
17.59 

917167 
916109 

56 

081709 

n 

29 

996S12  : 

26 

084947 

17-55 

9I3053   4  1 

57 

°IlMl 

17 

20 

996797  ; 

26 

086000 

i7-5r 

914000 

3 

58 

17 

21 

9967^2  ! 

26 

087050 

;?:s 

912950 

2 

59 

084864 

17 

17 

996766  j 

2b 

0S8098 
089144 

91 1902 
910856 

I 

60 

■ 

1 

0S5894 

17.13 

996-51 

26 

17-38 

0 

,  Cosine 

D. 

Sin- 

(-'or^n^:. 

D. 

Tang,  i  M.  | 

(83    DEGREES.) 


SINES   AND   TANGENTS.       (7   DEGREES.) 


25 


M. 

o 

Sine      D. 

Cosine 

D. 

Tang. 

1   D. 

1 

Cotang.  1 

9-085894   17 

•  i3 

9-9q675i 

.26 

9-089144 

17-38 

10.910856  60 

I 

086922   17 

-09 

996735 

.26 

090187 

17-34 

909813   5o 
90S772   58 

2 

087947   17 

.04 

996720 

.26 

091228 

I7-30 

3 

088970   17 

-00 

996704 

.26 

092266 

17.27 

907734   57 

4 

089990  !  16 

.96 

996688 

.26 

093302 

17-22 

906698   56 

5 

091008  1  16 

•  92 
-88 

996673 

.26 

094336 

17-19 

903664   55 

6 

092024  !  16 

996657 

.26 

095367 

i7-n 

904633   54 

I 

093037  1  16 

.84 

996641  1 

•  26 

096395 

17-11 

9o36o5   53 

094047  1  16 

80 

996625 

•  26 

097422 

17.07 

902578 

52 

9 

095o56   16 

76 

9966 1 0 

.26 

09S446 

17-03 

901554 

5i 

10 

096062   16 

73 

996594 

.26 

099468 

16-99 

900532 

5o 

II 

9-097065   16 

68 

9-996578 

•27 

9-100487 

16-95 

10.899513 

1? 

12 

098066   16 

65 

996562 

•27 

ioi5o4 

16-91 

898496 
897481 

i3 

099065   16 

61 

996546 

•27 

102319 

16-87 

47 

U 

^  100062    16 

57 

996530 

•27 

io3532 

16.84 

896468 

46 

i5 

ioio56   16 

53 

996514 

.27 

104542 

16.80 

895458 

45 

i6 

102048   16 

$ 

996498 

•27 

io555o 

16.76 

894450 

44 

\l 

io3o37   16 

996482 

•27 

106556 

16.72 

893444 

43 

io4o25   16 

41 

996465 

•27 

107559 

16.69 
16. 6d 

892441 

42 

19 

loSoio   16 

38 

996449 

•27 

io856o 

891440 

41 

20 

105992   16 

34 

996433 

•27 

109559 

16. 6i 

890441 

40 

21 

9-106973   16 

3o 

9-996417 

•27 

9-110556 

16.58 

10-889444 
888449 

39 

22 

107931    16 

27 

996400 

•  27 

iii55i 

16.54 

38 

23 

108927   16 

23 

996384 

•27 

112543 

16. 5o 

887457 

ll 

24 

1 0990 1    16 
1 10873   16 

19 

996368 

•27 

113533 

16-46 

886467 

25 

16 

996351 

•27 

1 1 452 1 

i6-43 

885479 

35 

26 

1 1 1842   16 

12 

996335 

•27 

ii55o7 

16.39 

884493 

34 

27 

1 1 2809   16 

08 

996318 

•27 

1 16491 

16.36 

883309 

33 

28 

113774   16 

o5 

996302 

.28 

117472 

16-32 

882328 

32 

29 

114737   16 

01 

996285 

.28 

118452 

16-29 
16-25 

881548 

3i 

3o 

11569S   1 5 

97 

996269 

.28 

1 19429 

880571 

3o 

3i 

9-II6656   i5 

94 

9-996232 

.28 

9-120404 

16-22 

'°i]& 

ll 

32 

1 1 761 3   1 5 

r, 

996235 

.28 

121377 

i6-i8 

33 

118567   i5 

996219  1 

.28 

12234S 

16. i5 

877632 

27 

34 

119519   i5 

83 

996202 

.28 

123317 

i6-ii 

8766S3  26 

35 

120469   1 5 

80 

996185 

.28 

124284 

16-07 

875716   25 

36 

1 2 1417   1 5 

76 

996168 

.28 

125249 

16-04 

874751  24 

u 

122362    l5 

73 

996151  1 

.28 

126211 

16-01 

872S28   22 

i233o6   i5 

69 

996134 

•  28 

127172 

i5-97 

39 

124248   1 5 

66 

996117 

•  28 

i28i3o 

i5-94 

871S7O   21 

40 

125187   i5 

62 

996100 

.28 

1 29087 

i5-9i 

870913   20 

4i 

9-126125   i5 

59 

9-996083 

•29 

9.130041 

15-87 

10-869959   19 
869006   18 

42 

127060   i5 

56 

996066 

•29 

1 30994 

15.84 

43 

127993   i5 
128925   i5 
129854   i5 

52 

996049 

-29 

131944 

i5.8i 

868036 

'7 

44 

49 

996032 

•29 

1 3 2^93 
133839 

15-77 

867107 

16 

45 

45 

996015 

•29 

i5-74 

866161   i5 

46 

130781   i5 

42 

995998 

•29 

134784 

i5-7i 

863216   14 

ii 

131706   i5 

U 

9959^0 

•29 

135726 

15-67 

864274   1 3 

i3263o   i5 

995963 

.29 

136667 

15-64 

863333   12 

49 

i3355i   i5 

32 

995946 

•29 

137605 

i5-6i 

862395   1 1 

5o 

134470   i5 

29 

995928 

•29 

138542 

i5-58 

861408   10 

5i 

9-135387   i5 

25 

9-995911 

995894 

•29 

9-139476 

i5-55 

io-86o524   9 
859591  i  8 

52 

i363o3   i5 

22 

•29 

1 40409 

i5.5i 

53 

137216   i5 

19 

995876 

•29 

141340 

15.48 

858660  1  7 

54 

i38i28   i5 

16 

995859 

•29 

142269 

15.45 

857731  1  6 

55 

139037   i5 

12 

995841 

•29 

143 196 

15.42 

8568o4   5 

56 

139944   1 5 
i4o85o   1 5 

09 

995823 

•29 

144121 

15-39 
15-35 

855879   4 

U 

06 

995806 

•  29 

145044 

854956  \     3 

141754   i5 

o3 

995788 

•  29 

145966 
146885 

15-32 

854034  1  2 

59 

142655   i5 

00 

99577  J 

•29 

i5-29 

853ii5  1  I 

66 

143555   14 

96 

995753 

•29 

147803 

i5-26 

852197  ;   0 

Cosine    D 

'   Sine   1 

Cotano:. 

D. 

Taug.   ' 

M.J 

(82    DEGREES.) 


26 

(8 

DEGREES.)   A  TABLE  OF  LOGARITHMIC 

M. 

Sine 

D. 

Cosine  j 

D. 

Tang. 

D. 

Cotang. 

o 

9.143555 

14-96 

9-995753 

.30 

9.147803 

15.26 

IO-852I97 

60 

I 

144453 

,  14-93 

995735 

•  3o 

148718 

15-23 

851282  1  59  1 
85o368  1  58  I 

2 

i4534q 

1  I4-0O 

1  14-87 

995717 

•30 

149632 

l5.20 

3 

146243 

995699 

•30 

i5o544 

15..7 

849456 

'^ 

4 

I47i36 
148026 

14-84 

995681 

•3o 

i5i454 

i5.i4 

848546 

5 

i4-8i 

995664 

.30 

152363 

i5.ii 

847637 

55 

6 

i48oi5 
149802 

14-78 

995646 

•3o 

153269 

i5.o8 

B46731 

54  ! 

I 

14-75 

995628 

•3o 

154174 

i5.o5 

845826 

53  1 

i5o686 

14-72 

995610 

•30 

155077 

l5.02 

844923  '  52  < 

9 

i5i569 

14-69 

995591 

-3o 

155978 

14-99 

844022  i  5l  ; 

10 

i5245i 

14-66 

995573 

•30 

156877 

14-96 

843123 

5o  1 

II 

9-153330 

14-63 

9-995555 

.30 

9.157775 

14-93 

10-842225 

49! 

12 

1 54208 

i4-6o 

995537 

-30 

158671 

14-87 

841329 

48 

i3 

i55o83 

14-57 

995519 

-3o 

159565 

840435  i  47 

U 

;^iS 

14-54 

995501 

•31 

160457 

14-84 

839543  ;  46  , 

838653  i  45  , 

i5 

i4-5i 

995482 

-3i 

i6i347 

i4-8i 

i6 

157700 

14-48 

995464 

-3i 

162236 

14-79 

837764  '  44  ; 

n 

158569 
159435 

14-45 

995446 

-3i 

i63i23 

14-76 

836877  !  43 

i8 

14-42 

995427 

-3i 

164008 

14-73 

835992  '  42 

19 

i6o3oi 

14-39 

995409 

•3i 

164892 

14-70 

835io8  !  41 

20 

161164 

14-36 

993390 

.31 

165774 

14-67 

834226  :  40 

21 

9-162025 

14-33 

9-995372 

.3i 

9.166654 

14-64 

10-833346  i  39 

832468  !  38  1 

22 

162885 

i4-3o 

995353 

.3i 

167532 

i4-6i 

23 

163743 

14-27 

995334 

.3i 

168409 

14-58 

83i59i  ■  37  i 

24 

164600 

14-24 

995316 

•  31 

169284 

14-55 

830716  36 

25 

165454 

14-22 

995297 

•  3i 

170157 

14-53 

829843   35 
828971   34 

26 

i663o7 

14-19 

995278 

.31 

171029 

14 -5o 

27 

\tia 

14-16 

995260 

.3i 

171899 

14-47 

828101   33 

28 

i4-i3 

995241 

.32 

172767 

14-44 

827233  i  32 

29 

168856 

14-10 

993222 

.32 

173634 

14-42 

826366  i  3i 

3o 

169702 

14-07 

995203 

.32 

174499 

14-39 

825501  ,  3o 

3i 

9-170547 

i4-o5 

9-995i84 

.32 

9.175362 

14-36 

10-824638   29 
823776   28 

32 

171389 

14-02 

995 I 65 

•32 

176224 

14-33 

33 

172230 

i3-99 

993146 

.32 

177084 

i4-3i 

822916  1  27 

34 

173070 

13-96 

993127 

.32 

177942 

14-28 

822058 

26 

35 

173908 

13-94 

995108 

.32 

178799 
179655 

14-25 

821201 

25 

36 

174744 

13-01 
13-88 

995089 

•32 

14-23 

820345 

24 

37 

175578 

995070 

•32 

i8o5o8 

14-20 

819492 
818640 

23 

38 

176411 

13-86 

995o5i 

•32 

i8i36o 

14-17 
14-15 

22 

39 

177242 

13-83 

O95o32 

•32 

182211 

817789  21  1 

40 

178072 

i3-8o 

9950 I 3 

•32 

i83o59 

14-12 

816941 

20 

41 

9-178900 

13-77 

9-994993 

.32 

9-183907 

14-09 

10-816093 

;? 

42 

179726 

i3-74 

994974 

•32 

184752 

14-07 

815248 

43 

i8o55i 

13-72 

994935 

•32 

i855o7 
186439 

14-04 

8i44o3 

n 

44 

181374 

13-69 

994935 

.32 

14-02 

8i356i 

16 

45 

182196 

13-66 

994916 
994896 

.33 

187280 

13-99 

8l2720 

i5 

46 

i83oi6 

13-64 

.33 

18S120 

13-96 

81 1880 

14 

% 

183834 

i3-6i 

994877 

.33 

188938 

13-93 

81 1042 

i3 

18465 1 

i3-59 
13-56 

994^57 

.33 

189794 

13-91 
13-89 

810206 

12 

49 

185466 

994838 

•  33 

190629 

809371    11  1 

5o 

186280 

13-53 

994818 

.33 

191462 

i3-86 

8o8538 

10 

5i 

9-187092 

i3-5i 

9.994798 

.33 

9-192294 

13-84 

10-807706 

t 

52 

'X:l 

13.48 

994779 

.33 

193124 

i3-8i 

806876 

53 

13-46 

994759 

.33 

193953 

13-79 

806047 

7 

54 

189519 
190325 

13.43 

994739 

.33 

194780 

13.76 

8o5220 

6 

55 

13-41 

994719 

.33 

193606 

13.74 

804394 

5 

56 

191130 

i3-38 

994700 

.33 

196430 

13.71 

803570 

4 

ll 

191933 

i3-36 

994680 

.33 

197253 

802747 

3 

192734 

13-33 

994660 

.33 

198074 

13-66 

801926 

2 

59 

193534 

13-30 

994640 

.33 

198894 

13-64 

801 106 

I 

60 

194332 

13-28 

994620 

.33 

199713 

i3-6i 

800287   0 

Cosine 

D.   i 

Sine   1 

Cotaucr. 

D. 

Tauff.  1  M, 

(81  D 

EGR 

KES.) 

SIXES   AND    TANGENTS.       (9    DEGREE.) 


27 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

i   D- 

Cotang. 

o 

9-194332 

i3-28 

9-994620 

.33 

9.199713 

i3-6i 

10-800287   60 

I 

195129 

13-26 

994600 

•  33 

200529 

i3 

-59 

799471  :  5o 
798655   58 

2 

195923 

13-23 

994580 

.33 

2oi345 

i3 

-56 

3 

196719 

13-21 

994560 

•34 

202159 

i3 

•54 

797841   57 

4 

197511 

i3-i8 

994540 

•34 

202971 

i3 

-52 

797029  56 
796218   55 

5 

19S302 

i3-i6 

994519 

•34 

203782 

i3 

•49 

6^ 

I 9909 I 

13-13 

994499 

34 

204592 

i3 

•47 

795408  !  54 

7^ 

199S79 

I3-II 

994479 

34 

205400 

i3 

•45 

794600   53 

8 

200666 

i3-o8 

994459 
994438 

34 

206207 

1 3 

•42 

793793   52 

9 

2oi45i 

i3-o6 

34 

207013 

i3 

-40 

7929S7   5i 

10 

202234 

i3-o4 

994418 

34 

207817 

i3 

38 

792183   5o 

u 

9-2o3oi7 

i3-oi 

9-994397 

34 

9.208619 

i3 

35 

10-791381  '  49 

12 

203797 

12-99 

994377 

34 

209420 

i3 

33 

7oo58o  1  48 

13 

204577 

12-96 

994357 

34 

210220 

i3 

3i 

789780  ]  47 
788982  1  46 

14 

205354 

12-94 

994336 

34 

211018 

i3 

28 

i5 

2o6i3i 

12-92 
12-89 

994316 

34 

2ii8i5 

i3 

26 

788185  '  45 

i6 

206906 

994295 

34 

212611 

i3 

24 

7873S9  ;  44 

•7 

207679 

12-87 

994274 

35 

2i34o5 

i3 

21 

786593  i  4i 

i8 

208452 

12-85 

994254 

35 

214198 

i3 

19 

785802   42 

'9 

209222 

12-82 

994233 

35 

2149^9 

i3 

17 

785011  j  41 

20 

209992 

12-80 

994212 

35 

215780 

i3 

i5 

784220  1  40 

21 

9-210760 

12.78 

9-994191 

35 

9-216568 

i3 

12 

10.783432  ;  39 
782644  1  38 

22 

2U526 

12.75 

994171 

35 

217356 

i3 

10 

23 

2I229I 

12-73 

994i5o 

35 

218142 

i3 

08 

781858  1  37 
781074  3o 

24 

2i3o55 

12-71 

994129 
994108 

35 

218926 

i3 

o5 

25 

2i38i8 

12-68 

35 

219710 

i3 

o3 

780290  ;  35 

26 

214579 

12-66 

994087 

35 

220492 

i3 

01 

779308  34 
778728  33 

27 

215338 

12-64 

994066 

35 

221272 

12 

99 

28 

216097 

12.61 

994045 

35 

222052 

12 

97 

777948   32 

29 

216854 

12.59 

994024 

35 

222830 

12 

94 

777170  3i 

3o 

2 1 7609 

12.57 

994oo3 

35 

2236o6 

12 

92 

776394  3o 

3i 

9-218363 

12-55 

9-993981 

35 

9-224382 

12 

9° 

10.775618   29 

774844  '   28 

32 

219116 

12.53 

993960 

35 

225i56 

12 

88 

33 

21Q868 

12. 5o 

993939 
993918 

35 

225929 

12 

86 

774071  1  27 

34 

220618 

12.48 

35 

226700 

12 

84 

773300  :  26 

35 

221367 

12.46 

993896 

36 

227471 

12 

81 

772529   25 

36 

222Il5 

12-44 

993S75 

36 

228239 

12 

79 

771761   24 

37 

222.S61 

12.42 

993854 

36 

229007 

12 

77 

770993  :  23 

38 

2236o6 

12.39 

993832 

36 

229773 

12 

75 

770227   22 

39 

224349 

12.37 

993811 

36 

23o539 

12 

73 

769461    21 

40 

225092 

12.35 

993789 

36 

23l302 

12 

71 

768698   20 

41 

9-225833 

12-33 

9-993768 

36 

9-232065 

12 

69 

10.767935  '    19 

42 

226573 

I2-3l 

993746 

36 

232826 

12 

67 

767174   18 

43 

227311 

12-28 

993725 

36 

233586 

12 

65 

766414   17 

44 

22S048 

12-26 

993703 

36 

234345 

12 

62 

765655   16 

45 

228784 

12-24 

993681 

36 

235io3 

12 

60 

764897   i5 

46 

229518 

12-22 

993660 

36 

235859 

12 

58 

764141  '  14 

47 

230252 

12-20 

993638 

36 

236614 

12 

56 

7633%   i3 

48 

2309S4 

12-l8 

993616 

36 

237368 

12 

54 

762632   12 

49 

231714 

12.16 

99  "5  594 

37 

238l20 

12 

52 

7618S0   11 

DO 

232444 

12-14 

993572 

37 

238872 

12 

5o 

761128   10 

5i 

9-233172 

12-12 

9-993550 

37 

9-23962^ 

12 

48 

10-760378  ;  9 
759629  !  8 

52 

2,33899 

12-09 

993528 

37 

240371 

12 

46 

53 

23462D 

12-07 

993506 

37 

241118 

12 

44 

7588«2  1  7 

54 

235349 
236073 

I2-o5 

993484 

37 

241865 

12 

42 

758,35   6 

55 

12-o3 

993462 

37 

242610 

12 

40 

7573QO   5 

56 

236795 

12-01 

993440 

37 

243354 

12 

38 

756646   4 

ll 

2375i5 

11-99 

993418  1 

37 

244097 

12 

36 

755903   3 

23S235 

11-97 

993396  i 

37 

244839 

12 

34 

755161  ,  2 

59 

23S953 

11-95 

993374 

37 

245579 

12 

32 

754421  1  1 

60 

239670 

11-93 

993351  j 

37 

246319 

12-3o 

753681 

- 

Cosine 

D. 

Sine 

Cotaiig. 

D. 

Tang. 

M.  1 

(80    DEGREES.) 


28 


(10    DEGREES.)      A   TABLE    OF    LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

D. 

Tang.     I 

).   1  Cotang.     1 

0 

9-239670 

1 1. 93 

9-993351 

V 

9-246319   12 

3o 

10-753681  1  60 

1 

240386 

II. 91 

993329 

37 

247057   12 

28 

752943   5o 
752206  58 

2 

241 lOI 

11.89 

993307 

37 

247794   12 
248530   12 

26 

3 

241814 

11.87 

993285 

37 

24 

751470  57 

4 

242326 

11-85 

993262 

37 

249264   12 

22 

750^36  i  56 

5 

243237 

11-83 

993240 

ll 

iw;il  \i 

20 

750002 

55 

6 

243947 

11-81 

993217 

18 

749270 

54 

I 

244656 

11-79 

993195 

38 

251461  12 

17 

748539 

53 

245363 

11-77 

993172 

38 

252191   12 

i5 

747809 

52 

9 

246069 

11-75 

993149 

38 

252920  12 

i3 

747080 

5i 

10 

24677D 

11-73 

993127 

38 

253648   12. 

II 

746352 

5o 

II 

9-247478 

11-71 

9-993104 

38 

9-254374   12 

09 

10.745626 

% 

12 

248 1 81 

11-69 

993081 

38 

255ioo   12 

07 

744900 

i3 

248883 

11-67 

993059 

38 

255824   12 

o5 

744176  1  47 
743453  '  46 

14 

249583 

11-65 

993 o3 6 

38 

25654/7   12 

o3 

ID 

250282 

11-63 

993oi3 

38 

257269   12 

01 

742731   45 

i6 

250980 

11-61 

992990 

38 

237990   12 
238710   II 

00 

742010  44 

17 

251677 

Vi'M 

992967 

38 

98 

741290  43 

i8 

252373 

992944 

38 

239429   II 

96 

740571  1  42 

19 

253067 

11.56 

992921 

38 

260146   II 

94 

739854 

41 

20 

253761 

11-54 

992898 

38 

260863   II 

92 

739137 

40 

21 

9 -.2  54453 

11.52 

9.992875 

38 

9-261578   11 

% 

10.738422 

3o 

22 

255i44 

11.50 

992852 

38 

262292   11 

737708  !  38  i 

23 

255834 

11.48 

992829 

39 

263oo5   1 1 

ll 

736995  !  37  1 

24 

256523 

11.46 

992806 

39 

2637.7   ,1 

736283  ;  36 

25 

257211 

11-44 

992783 

39 

264428   1 1 

83 

735572.  1  35 

26 

257898 

11-42 

992759 

39 

265i38   II 

81 

734862  1  34 

27 

253583 

11-41 

992736 

39 

265847   II 

]t 

734153  1  33 

28 

25926S 

11.39 

992713 

39 

266555   11 

733445  1  32 

29 

209951 

11.37 

992690 

39 

267261    11 

76 

732739  3i 
732033  3o 

3o 

260633 

11-35 

992666 

39 

267967   11 

74 

3i 

9-26i3i4 

11-33 

9.992643 

39 

9-268671    II 

72 

io-73i329   29 
730625   28 

32 

261994 

ii-3i 

992619 

39 

269375   n 

70 

33 

262673 

II-30 

992596 

39 

270077   1 1 

69 

729923  j  27 

34 

263351 

11-28 

992572 

39 

270779   II 

67 

729221  1  26 

3d 

264027 

11-26 

992349 
992525 

39 

271479   II 

65 

728521  1  25 

36 

264703 

11-24 

39 

272178   II 

64 

•727822  1  24  i 

,37 

265377 

11-22 

992501 

39 

272876   II 

62 

727124  j  23 

38 

26605 I 

11-20 

992478 

40 

273573   II 

60 

726427   22 

•  39 

266723 

11-19 

992454 

40 

274269   II 

58^ 

725731    21 

40 

267395 

11-17 

992430 

40 

274964   11 

57 

725o36   20 

41 

9.268065 

II. i5 

9.992406 

40 

9-275658   11 

55 

10.724342  !  19 
723649   18 

42 

268734 

II. i3 

9923^2 

40 

276351   II 

53 

43 

269402 

ii-ii 

992359 

40 

277043   11 

5i 

722957   17 

44 

270069 

II. 10 

992335 

40 

277734   II 

5o 

722266   16 

45 

270733 

11.08 

9923 1 1 

40 

278424   II 

48 

721376  j  i5 

46 

271400 

11.06 

992287 

40 

279«i3   II 

47 

720887  1  14 

% 

272064 

11. o5 

992263 

40 

279801    II 

45 

720199   i3 

272726 

11-03 

992239 

40 

280488   1 1 

43 

719312   12 

49 

273388 

II. 01 

992214 

40 

281174   II 

41 

718826   II 

5o 

274049 

10-99 

992190 

40 

28i858   II 

40 

718142   10  1 

5i 

9-274708 

10-98 

9-992166 

40 

9.282542   II 

38 

10.717438  ;   9 

716775  ;  8  i 

52 

275367 

10-96 

992142 

40 

283225    II 

36 

53 

276024 

10-94 

992117 

41 

284588   1 1 

35 

716093  !  7 

54 

276681 

10-92 

992093 

41 

33 

715412  i  6 

55 

277337 

10.91 
10-89 

992069 

41 

285268   II 

3i 

714732  :  5 

56 

277991 

992044 

41 

285947   1 1 

30 

714053  :  4 

57 

278644 

10-87 
10-86 

992020 

41. 

286624   1 1 

28 

713376  :  3 

58 

279297 
27994B 
280399 

991996 

41 

287301    II 

26 

712699   2 

712023  :   I 

59 

10.84 

991971 

41 

287977   1 1 

25 

60 

10-82 

991947  1 

41 

288652   1 1 

23 

711348  ,   0 

Co&iiie 

D. 

Sine 

Cotiing.     D 

i  Tang.   1  M.  | 

(79  DEGREES.') 


SINES   AKD   TANGENTS.      (11    DEGREES.) 


29 


0 

Sine 

D. 

Cosine 

D. 

Tang. 

D.   ! 

Cotang. 

9-280599 

10-82 

9 -9c;  1 947 

•41 

9-288652 

11-23 

10^71 1348  60 

I 

28.248 

10 

81 

991922 

■41 

289826 

11-22 

710674  ;  5q 

71000.   58 

2 

281897 

10 

79 

991897 

•41 

289999 

11-20 

3 

282544 

10 

77 

99.873 

•41 

290671 

11-18 

709829  1  57 

4 

283190 

10 

76 

991848 

•41 

29.342 

11-17 

708658 

56 

5 

283836 

10 

74 

991823 

•  41 

2920.3 

II-15 

707987 

55 

6 

284480 

10 

72 

991799 

•41 

2926S2 

11-14 

707818 

54 

I 

285124 

10 

11 

991774 

•42 

293350 

11-12 

7066 5o 

53 

285766 

10 

69 

991749 

.42 

294017 

II  -11 

700983 

52 

9 

286408 

10 

67 

991724 

•42 

294684 

11-09 

7o53i6 

5i 

10 

287048 

10 

66 

99 1 099 

•42 

295349 

11^07 

704601 

5o 

II 

9.287687 

10 

64 

9-991674 

•42 

9-2960.3 

11-06 

10-703987 

49 

12 

28S326 

10 

63 

991^^+9 

•42 

296677 

11-04 

703 J23 

48 

1 3 

288964 

10 

61 

991^^24 

•42 

297339 

ii-o3 

702661 

47 

14 

289600 

10 

59 

99.599 

•42 

298001 

11  -01 

701999 
701338 

46 

i5 

290236 

10 

58 

99.574 

•42 

298662 

11-00 

45 

i6 

290870 

10 

56 

991549 

•42 

299822 

10-98 

700678 

44 

17 

29i5o4 

10 

54 

99.524 

•42 

299980 

10-96 

700020 

43 

i8 

292137 

10 

53 

99.49^ 

•42 

3oo638 

10-95 

699862 
698705 

42 

'9 

292768 

10 

5i 

99.473 

•42 

301295 

10-98 

41 

20 

293399 

10 

5o 

991448 

•42 

30.95. 

10-92 

698049 

40 

21 

9-294029 

10 

48 

9-99.422 

•42 

9-302607 

10-90 

10-89 

10-697398 

ll 

22 

294658 

10 

46 

991897 

•42 

3o326i 

696789 

23 

295286 

10 

45 

991372 

•43 

303914 

10-87 
10-86 

696086 

ll 

24 

295qi3 

10 

43 

991346 

•43 

304567 

695433 

25 

296539 

10 

42 

99.32  1 

•43 

3o52i8 

10-84 

6947''52 

35 

26 

397 1 '^4 

10 

40 

991295 

•43 

3o5%9 

10-83 

69418! 

34 

27 

297788 

10 

39 

99.270 

•43 

3o65i9 
307.68 

10-81 

693481 

33 

28 

298412 

10 

37 

991244 

•43 

io-8o 

692882 

32 

29 

299034 

10 

36 

991218 

•43 

3078.5 

10-78 

692185 

3i 

3o 

299655 

10 

34 

99.193 

•43 

3o8463 

10^77 

691537 

So 

3i 

9-300276 

10 

32 

9-99.167 

•43 

9 '309 1 09 

10^75 

10-690891 

li 

32 

300895 

10 

3i 

99'i4i 

•43 

309754 

10^74 

690246 
689602 

33 

3oi5i4 

10 

li 

991115 

•43 

3.0898 

10-73 

27 

34 

302l32 

10 

991090 

•43 

3.1042 

10-71 

688958 

26 

35 

302748 

10 

26 

99.064 

•43 

3ii685 

10-70 

6883.5 

25 

36 

3 03354 

10 

25 

991038 

•43 

3.2827 

10-68 

687673 

24 

ll 

3o397Q 

10 

23 

9910.2 

•43 

3.2967 

10-67 

687033 

23 

304593 

10 

22 

990986 

•43 

3\36o8 

10-65 

686892 

22 

39 

3o5207 

10 

20 

990960 

.43 

3.4247 

10-64 

685753 

21 

40 

3o58i9 

10 

19 

990934 

•44 

3.4885 

10-62 

6851.5 

20 

41 

9-3o643o 

10 

17 

9-990008 
990882 

•44 

9^3. 5523 

10-61 

10-684477 

\l 

42 

307041 

10 

16 

•44 

3.6.59 

10-60 

683841 

43 

3o;65o 

10 

14 

990855 

•44 

816790 

10-58 

683205 

17 

44 

308209 

10 

i3 

990829 

•44 

817480 

10-57 

682570 

16 

45 

308867 

10 

11 

990803 

•44 

818064 

10-55 

68.986 
681803 

i5 

46 

309474 

10 

10 

990777 

•44 

1   3.8697 

10-54 

14 

47 

3 1 0080 

10 

08 

990750 

•44 

3.9329 

10-53 

680671 

i3 

48 

3io685 

10 

07 

990724 

•44 

'   3.9961 

io-5i 

68oo3n 

679408 

13 

49 

311289 
311893 

10 

o5 

990697 

•44 

320392 

io^5o 

11 

5o 

10 

04 

990671 

•44 

i    321222 

10-48 

678778 

10 

5i 

9-812495 

10 

o3 

9  990644 

•44 

'  9-32.851 

10-47 

10-678149 

? 

52 

3.3097 

10 

01 

9906.8 

•44 

322479 

10-45 

677521 

53 

313698 

10 

00 

990591 

•44 

328106 

10-44 

676894 

I 

54 

3.4297 

9 

98 

990565 

•44 

;  323733 

10-43 

676267 

55 

3.4897 

9 

97 

990538 

•44 

i   324353 

10-41 

675642 

5 

56 

315495 

9 

96 

9905 11 

•45 

!  3249S3 

10-40 

675017 

4 

57 

316092 
316689 

9 

94 

9904 -i;i 

•45 

i   325607 

10-89 

674898 

3 

58 

9 

93 

9904)8 

•45 

1   32628. 

10-37 

673769 

2 

59 

317284- 

9 

91 

99043. 

•45 

1   326853 

10-36 

673147 

I 

60 

317879 

9-90 

9904  '4 

•45 

'   327475 

10-35 

672525  •  0  1 

Cosine 

D. 

Sine 

i  Cotans-. 

D. 

Tans-  IJIL 

(Y8    DEGREES.) 


80 


(12    DEGREES.)      A  TABLE   OF   LOGARITHMIC 


JM.| 

Sine 

D.   1 

Cosine  1 

D. 

Tang. 

D. 

Cotang.  1    1 

o 

9.317879 

9.90 

9-9Q0404 

•45 

9.327474 

'°"35 

10-672526  i  60 

1 

318473 

9-88 

990378 

•45 

328093 

10-33 

671903  ;  59 
671285   58 

2 

319066 

9-87 

99035 1 

•45 

3287.5 

10-32 

3 

319658 

9-86 

990324 

•45 

329334 

10.30 

670666  ;  57 

4 
5 

320249 
320840 

9.84 
9-83 

990297 
990270  [ 

•45 
•45 

329933 
33o5io 

10.29 
10.28 

670047  ;  56 
669430  i  55 

66881 3   54 

6 

321430 

9-82 

990243  1 

•45 

33.. 87 

10-26 

7 

322019 

9-80 

990215  1 

•45 

33.8o3 

10-25 

668197   53 

8 

322607 

9-79 

990188 

•45 

332418 

10.24 

6675M2 

52 

9 

323io4 

9-77 

990161 

•43 

333o33 

10-23 

666967 

5i 

10 

323780 

9-76 

990134 

•45 

333646 

10.21 

5o 

II 

9-324366 

9.75 

9-990107 

•46 

9-334239 

10.20 

10 -665141 

it 

12 

324950 

9.73 

990079 

.46 

334871 

10.19 

665 i 29 

i3 

325334 

9-72  ■ 

990052 

•46 

3354«2 

10.17 

664518 

47 

U 

326117 

9.70 

990025 
9S9997 

.46 

336093 

10.16 

663907 

46 

ID 

326700 

III 

•46 

336702 

I0.I5 

663298 
662689 

45 

i6 

327281 

989070 

•46 

3373.1 

10.13 

44 

]l 

327862 

9-66 

9S9942 

.46 

337919 

10-12 

6620S1 

43 

328442 

9-65 

l^l^ 

•46 

338327 

lO-II 

661473  42  1 

'9 

329021 

9.64 

•46 

339133 

10.10 

660867 

41 

20 

329599 

9-62 

989860 

.46 

33973^ 

10.08 

660261 

40 

21 

9-330176 

9-6i 

9.989832 

.46 

9-340344 

10.07 

10.639656  39 
659032   38 

■  22 

330753 

9-60 

989804 

.46 

340948 

10.06 

23 

33i329 

9-58 

9^^9777 

•46 

341332 

10.04 

658448  ;  37 

657845  :  36 

24 

331903 

9-57 

989749 

•47 

342r55 

io-o3 

25 

332478 

9-56 

989721 

•47 

342757 
343358 

10-02 

657243  '  35 

26 

^   333o5i 

9-54 

989693 

•47 

10-00 

656642   34 

27 

333624 

9-53 

9S9665 

•47 

343958 

9. 90 
9-98 

636042   33 

28 

334195 

9-52 

989637 

•47 

344558 

655442   32 

29 

334766 

9.50 

989609 

•47 

343157 

9-97 

.654843   3i 

3o 

335337 

9.49 

989382 

•47 

345755 

9.96 

654245   3o 

3i 

9-335906 

9-48 

9-989553 

•47 

9-346353 

9-94 

10-653647   29 
653o5i   28 

32 

336475 

9.46 

989525 

•47 

346949 

9-93 

33 

337043 

9-45 

989497 

■47 

34-345 

9-92 

652455   27 

34 

337610 

9.44 

989469 

•47 

348141 

9.91 

65.859  '  26 

35 

338176 

9-43 

989441 

•47 

348735 

l^ 

65.263  ■  25 

36 

338742 

9-41 

989413 

•47 

349329 

650671   24 

11 

339305 

9.40 

989384 

•47 

349922 

III 

6500-8  1  23 

359871 

9.39 

gSt)356 

•47 

33o5i4 

6404%  ;  22 

39 

340434 

9-37 

Q89328 

•47 

35^106 

9-85 

648894  ■  21 

40 

340996 

9-36 

989300 

•47 

351697 

9.83 

6483o3   20 

41 

9-341558 

9-35 

9-989271 

•47 

9-332287 

9-82 

io-64'77.3   19 

42 

342119 

9-34 

9S9243 

•47 

352876 

9.81 

647  J  24 

18 

43 

342679 

9-32 

989214 

•47 

353465 

9-80 

646535 

n 

44 

343239 

9-3. 

989186 

•47 

354053 

9-79 

645947 

16 

45 

343797 

9-30 

989157 

:% 

354640 

9-77 

645360 

i5 

46 

344355 

9-29 

989128 

355227 

9-76 

6447^3 

.4 

tl 

344912 

9-27 

989100 

.48 

3558i3 

9-75 

644 I S7 

i3 

345469 

9-26 

989071 

•  48 

356398 
356982 

9-74 

643602 

12 

49 

346024 

9-25 

989042 

.48 

9-73 

643018   11  i 

5o 

346579 

9-24 

989014 

.48 

357366 

9.71 

642434 

10 

5i 

9-347134 

9-22 

9.988983 

.48 

9-358149 

9.70 

io.64i85i 

t 

52 

347687 

9-21 

98S956 

•  48 

35873. 

lU 

641269 

53 

348240 

9-20 

98S927 

.48 

3593.3 

640687 

7 

54 

i   348792 

9-19 

98S898 

.48 

359893 

9.67 

640107 

6 

55 

349343 

9-17 

9S8869 

.48 

360474 

9.66 

639326 

5 

56 

349^^93 

g-16 

988840 

•48 

36.053 

9.65 

63 8047 

4 

57 

;   350443 

9-15 

9888.1 

.49 

36.632 

9.63 

638368  ■  3 

58 

:  350992 

9-14 

988782  1 

•49 

362210 

9.62 

.  "7790   -2 

59 

i   35ID40 

9-i3 

988753  1 

•49 

362787 

9.61 

63-2.3   1 

1  60 

3520SS 

911 

988724  ; 

•49 

363364 

9-60 

636636  t  0  1 

1 

Copine 

i   D. 

Sine   i 

Cotan?. 

D. 

Taiig.  1  M.  i 

(77  DEGREES.) 


«INES   AND   TANGENTS.       (13    DEGREES.) 


81 


M. 

'   Sine 

D. 

Cosine 

D. 

Tan,. 

D. 

j  Cotang.  '    ] 

0 

9.352088 

9-II 

9-9-^8724 

.49 

9-3^3364 

9.60 

10-636636  i  60 

I 

352635 

9 

ID 

988695 

•49 

3639  to 

9.59 

636o6o  1  5o 

2 

353181 

9 

09 

9SS')66 

•49 

364315 

'   9^58 

635485  i  58 

3 

353726 

9 

•08 

98S6i6 

-49 

365090 

'   9-57 

634910  i  57 

4 

3)4271 

9 

•07 

98%:,7 

•49 

365664 

9-55 

634336  1  56 

5 

3 54s 1 5 

9 

•  o5 

988578 

•49 

366237 

9-54 

633763  !  55 

6 

355353 

9 

.04 

98854S 

•49 

366810 

9-53 

633190  !  54 

7 

355901 

9 

o3 

988519 

•49 

i   367382 

9-52 

632618  1  53 

8 

356443 

9 

•02 

9884S9 

•49 

367953 

9-5i 

632047  !  52 

9 

356984 

■01 

988460 

•49 

368524 

9-5o 

631476  '  5i 

10 

357524 

8 

99 

988430 

•49 

369094 

9.49 

630906  1  5o 

II 

9.358064 

8 

93 

9.988401 

•49 

9^369663 

9-48 

io-63o337   49 
62976S  48 

12 

3586o3 

8 

97 

98  S3  7 1 

•49 

370232 

9-46 

i3 

359141 

8 

96 

988342 

•49 

370799 

9-45 

629201  ;  47 

628633  1  46 

14 

359678 

8 

95 

9S.S3 1 2 

-5o 

371367 

9-44 

i5 

36021 5 

8 

93 

9SS2S2 

-5o 

371933 

9-43 

628067  j  45 

i6 

360752 

8 

92 

988212 

-50 

3-2 io9 

9-42 

627501  1  44 

\l 

361287 

8 

91 

988223 

.5o 

373064 

9-41 

626936  1  43 
626371  1  42 

361822 

8 

988193 

-5o 

373629 

9-40 

'9 

362  356 

8 

89 

988163 

-5o 

374193 

9-30 

625807  :  41 

20 

362% 

8 

88 

988133 

•5o 

374756 

9-3^ 

625244  ;  40  • 

21 

9-363422 

8 

87 

9-988103 

•So 

9-375319 

9-37 

10-624681  j  39 
624119  !  38 

22 

363954 

8 

85 

98S0-3 

•5o 

3758S1 

9-35 

23 

364485 

8 

84 

988043 

-50 

376442 

9-34 

623558  ■  37 

24 

36501 6 

8 

83 

988013 

.5o 

377003 

9-33 

622907  ;  36 

25 

3j5546 

8 

82 

987983 

.5o 

377563 

9-32 

622437  j  35 

26 

366075 

8 

81 

987953 

.50 

378122 

9-3i 

621878  1  34 

27 

366604 

8 

80 

987,22 

•  Do 

378681 

9-3o 

621319  1*33 

28 

367i3i 

8 

79 

987892 

-So 

379239 

9-29 
9-28 

620761  1  32 

^9 

367659 

8 

77 

9S7  ^62 

.5o 

379797 

620203  '  3i 

3o 

368 1 85 

8 

76 

987832 

•  51 

380354 

9.27. 

619646  I  So 

3i 

9.368711 

8 

75 

9-987801 

-5r 

9^380910 

9-26 

10-619090   29 
6 1 8534  '■   28 

32 

369236 

8 

74 

9^7771 

•  5i 

3Si466 

9-25 

33 

369761 

8 

73 

987740 

.5i 

382020 

9-24 

617980  :  27 
617425  i  26 

34 

370285 

8 

72 

987710 

.5i 

382575 

9-23 

35 

370808 

8 

71 

987679 

•  51 

383129 

9-22 

616871  '   25 

36 

37i33o 

8 

70 

9876*9 

•  51 

383682 

9-21 

6i63i8   24 

ll 

371852 

8 

69 

987618 

.5i 

384234 

9-20 

615766  '  23 

372373 

8 

tl 

987588 

-5i 

384786 

9-19 

6i52i4  ''   22 

39 

372^94 

8 

987557 

-5i 

383337 
385883 

9-18 

614663   21 

40 

373414 

8 

65 

987526 

.5i 

9-17 

614112  ;  20 

41 

9-373933 

8 

64 

9-937496 

-5i 

9^386438 

9-i5 

io-6i3562  '■■   19 
6i3oi3   18 

42 

374452 

8 

63 

987465 

-51 

386987 

9-14 

43 

3^970 

8 

62 

987434 

.5i 

387336 

9.. 3 

612464   17 

44 

375487 

8 

61 

987403 

-52 

388084 

9-12 

611916   i6 

45 

37600.3 

8 

60 

98757-^ 

.52 

388631 

9-11 

611369   1 5 

46 

376519 

8 

59 

987341 

.52 

389178 

9-10 

610822   14 

^I 

377035 

8 

58 

9.87310 

.52 

389724 

9.09 

610276   i3 

48 

377549 

8 

57 

987279 

-52 

390270 

9-08 

609730 

12 

49 

378063 

8 

56 

987 '^48 

-52 

390815 

9-07 

609185 

II 

5o 

.378577 

8 

•54 

987217 

.52 

391360 

906 

608640 

10 

5i 

9.379089 

8. 

53 

9-987186 

-52 

9.391903 

9-o5 

10-608097 

t 

52 

379601 

8 

52 

987155 

•52 

392  i47 

9  •  04 

607533 

53 

38oii3 

8- 

5i 

987 '24 

-52 

llltf, 

9-o3 

60701 1 

7 

54 

380624 

8 

5o 

987092 

-52 

9.02 

606469 

6 

55 

38ii34 

8- 

% 

987061 

-52 

394073 

9-01 

605937 

5 

56 

381643 

8 

987030 

-52 

394614 

9-00 
8-99 

6o5386 

4 

ll 

382x52 

8- 

47 

986998 

.52 

395154 

604846 

3 

382661 

8 

46 

986967 

•52 

395694 

8-98 

604306 

2 

59 

383 168 

8 

45 

9869  ]6 

-52 

396233 

1:'^ 

603767   I 

60 

383675 

8-44 

986904 

•52 

396771 

603229  1  0 

1 

Cofiine 

D. 

Sine 

Cotaua:. 

D. 

Tun?. 

M. 

26 

1 

(7G 

DECK 

EES.) 

32 


.4    DEGREES.)       A   TABLE    OF    LOGARITH^^lIC 


|m; 

Sine 

D. 

Cosine  ' 

D. 

T;mg. 

D. 

Cotaiig.  j 

0 

9-383675 

8-44 

9-986904 

52 

9.396771 

8-96 

10-603229  i  60 

1  I 

384182 

8-43 

9S6873 

53 

397309 

8 

96 

602691   So 
602104  58 

1   2 

384687 

8-42 

986841  ' 

53 

397846 

8 

90 

1   3 

380192 

8-41 

986809  1 

53 

398383 

8 

94 

601617   57 

i  4 

385697 

8.40 

986778 

53 

398919 

8 

93 

601081  ;  56 

1  5 

386201 

8-39 
8.3S 

9S6746  ; 

53 

399455 

8 

92 

600545  '  55 

'  6 

336704 

9867.4  ; 

53 

399990 

8 

9« 

600010  !  54  1 

I 

387207 

8-37 

986683  1 

53 

400024 

8 

i 

599476  1  53  i 

387709 

8-36 

986601  1 

53 

401058 

8 

598942  :  52 

9 

338210 

8-35 

9S6619  1 

53 

401091 

8 

598409  ,  01 

597876  !  5o 

10 

3887 II 

8-34 

9865S7  1 

53 

402124 

8 

87 

II 

9.3S9211 

8-33 

9 -986555  1 

53 

9-402606 

8 

86 

10.597344  !  49 
596S13  •  48 

12 

^389711 

8-32 

9%523  ! 

53 

4o3 1 87 

8 

85 

i3 

390210 

8.3i 

9^^6491  1 

53 

403718 

8 

84 

5962S2  '  47 

14 

390708 

8-3o 

9%459 

53 

404249 
404778 

8 

83 

595751  i  46 

i5 

391206 

8-28 

9S6427 

53 

8 

82 

595222  '  45 

i6 

391703 

8.27 

9S6395 

53 

4o53o3 

8 

81 

594692  :  44 

\l 

392199 

8-26 

986363 

54 

405836 

8 

80 

594164  .   43 

392695 

8-25 

986331 

54 

406364 

8 

?I 

593636  42 

19 

393191  • 

8-24 

986299  1 

54 

406892 

8 

593108  !  41 

.2p 

393685 

8-23 

986266 

54 

407419 

8 

77 

59258,  1  40 

21 

9-394179 

8-22 

9-986234 

54 

9-407945 

8 

76 

10-592005  !  39 

22 

394673 

8-21 

9S6202 

54 

408471 

8 

75 

591520  1  38  1 

23 

395166 

8-20 

986169 

54 

408997 

8 

74 

591003  i  37  1 

24 

395658 

8-IO 
8-18 

986137 

54 

409021 

8 

^ 

590479  i  36  j 
589950  I  35 

25 

396 I 5o 

986104 

54 

410045 

8 

73 

26 

396641 

8.17 

986072 

54 

410069 

8 

72 

58043 1   34 

^r 

397132 

8-17 
8-16 

986039 

54 

411092 

8 

71 

397621 

986007 

54 

4ii6i5 

8 

70 

588385  i  32  ! 

29 

3981 1 1 

8-i5 

980974 

54 

412137 

8 

^ 

587863  !  3i  i 

3o 

3986QO 

8.,4 

935942 

54 

412658 

8 

587342  j  3o  : 

3i 

9-399088 

8-i3 

't^s 

55 

9-4i3i79 

8 

67 

10-586821   29  ; 
586301  j  28  1 

32 

399575 

8-12 

55 

413699 

8 

66 

33 

400062 

8-11 

985843 

55 

414219 

8 

65 

535781 

,27  i 

34 

400549 

8-10 

98581 1 

55 

414738 

8 

64 

585262 

'26 

35 

401033 

8-09 
S-oS 

985778 

55 

415257 

8 

64 

584743 

25  ! 

36 

401 520 

985745 

55 

■  415775 

8 

63 

584225 

24  1 

37 

4o2oo5 

8-07 

985712 

55 

416293 

8 

62 

583707 

23 

38 

402489 

8-06 

985679 

55 

416810 

8 

61 

583 190 

22 

39 

402972 

8-o5 

985646 

55 

417326 

8 

60 

582674  21  1 

40 

403455 

8-04 

9856i3 

55 

417842 

8 

59 

582158 

20 

41 

9-403938 

8-03 

9-985580 

55 

9-418358 

8 

58 

i3-58i642 

:? 

42 

404420 

8-02 

9S5547 

55 

418873 

8 

57 

581127 

43 

4o4qoi 

8-01 

985514 

55 

4193S7 

8 

56 

58061 3 

17 

44 

4o5382 

8-00 

985480 

55 

419901 

8 

55 

580009 

16 

45 

4o5862 

V^ 

9*554-17 

55 

420415 

8 

55 

579585 

i5 

46 

406341 

935414 

56 

420927 

8 

54 

570073 

i4 

47 

406820 

7-97 

985380 

56 

421440 

8 

53 

578560   i3  1 

48 

407299 

7-96 

985347 

56 

421952 

8 

52 

578048   12  1 

49 

407777 

7-95 

980314 

56 

422463 

8 

5i 

577537  ,  11  ! 

5o 

408254 

7-94 

9S52S0 

56 

422974 

8 

5o 

577026  ;  10 

1  5i 

9-408731 

7-94 

9-985247 

56 

9. 4^484 

8 

^? 

10-576516  i  9 
576007  i  8 

'  52 

409207 

7-93 

985213 

56 

423993 

8 

53 

409682 

7-92 

985180 

56 

4245o3 

8 

48 

575497 

7 

54 

410157 

7.91 

985146 

56 

425oii 

8 

47 

5749''9 

6 

,  55 

4io632 

985ii3 

56 

425519 

8 

46 

574481 

5 

56 

41 1 106 

7.89 

985079 

56 

426027 

8 

45 

5739^3 

4  i 

U 

411579 

7 -So 

980043 

56 

426534 

8 

44 

573466  !  3  1 

4I2052 

7-87 

98501 1 

56 

427041 

8 

43 

5729^.0  I  2 
572453  1  I 

^9 

412524 

?:^ 

984978 

56 

427547 

8 

43 

60 

412996 

984944  ! 

56 

428052 

8-42 

571948  j  0  j 

1 

Cosine 

1   D. 

Sine 

Cotang. 

D. 

Ttnff.  !  M.  i 

(75  DEGREES.) 


SINES   AND   TANGENTS.      (15   DEGREES.) 


M. 

Sine 

D. 

Cosine  | 

D. 

Tan,?. 

D. 

Cotau-,'.  I 

0 

9-412996 

7-85 

9 -9849^4 

•57 

9-428002  . 

8.42 

10-571948  60 

I 

413467 

84 

9849 10 

•57 

428507 

8 

41 

571443   59 

2 

413933 

83 

984876 

•57 

429062 

8 

40 

570933  j  53 

3 

414408 

7 

83 

984S42 

•57 

429066 

8 

39 

570434  !  57 

4 

414B78 

82 

984808  1 

•57 

430070 

8 

38 

569930 

56 

5 

415347 
4i58i3 

7 

81 

984774  I 

•57 

430573 

8 

33 

56?925 

55 

6 

7 

80 

984740 

•57 

431075 

8 

37 

54 

I 

416283 

79 

984706  1 

-57- 

43 1 577 

8 

36 

568423 

53 

416731 

73 

984672 

1^ 

432079 

8 

35 

567921 

52 

9 

417217 

77 

9S4637 

•57 

432080 

8 

34 

567420  1  5i 

10 

417684 

76 

984603 

•57 

433080 

8 

33 

566920  5a 

II 

9-4i8i5o 

75 

9-984569  I 

•57 

9-433580 

8 

32 

lo- 566420  1  49 
565920  1  43 

12 

4j86i5 

74 

9S4535 

•57 

434080 

8 

32 

i3 

419079 

73 

984000 

•57 

434579 
430078 

8 

3i 

565421  i  47 

14 

419544 

73 

984466  1 

-57 

8 

3o 

564922  1  46 

i5 

420007 

72 

984432 

•  58 

435576 

8 

29 
28 

564424 

45 

i6 

420470 

71 

984397 

-58 

436073 

8 

563927 

U 

n 

420933 
42.395 

70 

984363 

-58 

436570 

8 

28 

563430 

43 

i8 

^? 

984328 

-58 

437067 

8 

ll 

562933 

42 

19 

421807 
4223i8 

984294 

-58 

437563 

8 

562437 

41 

20 

67 

984259 

•  58 

438009 

8 

25 

56I94I 

40 

21 

9-422778 

67 

9-984224  1 

.58 

9-438554 

8 

24 

10-561446 

39 

22 

423238 

66. 

984190  ! 

-58 

439048 

8 

23 

560952 

33 

23 

423697 
424156 

65 

984155 

.58 

439043 

8 

23 

560457 

37 

24 

64 

984120 

.58 

44oo36 

8 

22 

559964 

36 

25 

4246i5 

63 

984085 

•  58 

440529 

8 

21 

559471 

35 

26 

425073 

62 

984000 

.58 

441022 

8 

20 

553978 

34 

27 

425530 

61 

984015 

.58 

44i5i4 

8 

19 

558486 

33 

28 

425987 

60 

983981 

.58 

442006 

8 

19 

557994 

32 

29 

426443 

60 

983946 

.58 

442497 

8 

18 

557003 

3i 

3o 

426899 

59 

9839 1 1 

-53 

4429S8 

8 

17 

557012 

3o 

3i 

9-427354 

58 

9-983875 

.58 

9-443479 

8 

16 

10-556521 

ll 

32 

427809 

57 

983840 

.59 

443968 

8 

16 

556o32 

33 

428263 

56 

9838o5 

.59 

444458 

8 

i5 

555542 

27 

34 

428717 

55 

983770 

.59 

444947 

8 

14 

555o53 

26 

35 

429170 

54 

983735 

-59 

445435 

8 

i3 

554565 

25 

36 

429623 

53 

983700 

.59 

445923 

8 

12 

554077 

24 

ll 

430075 

52 

983664 

-59 

4464 1 1 

8 

12 

553539 

23 

43o527 
430978 

52 

983629 

.59 

446898 

8 

II 

553102 

22 

39 

5i 

983594 

.59 

447384 

8 

10 

5526.6 

21 

40 

431429 

5o 

983508 

.59 

447870 

8 

09 

552i3o 

20 

41 

9-431879 

49 

9-983523 

.59 

9-448356 

8 

2 

10-55.644 

19 

42 

432329 
432778 

49 

983487 

.59 

448841 

8 

55.109 

18 

43 

7 

48 

983452 

.59 

449326 

8 

07 

550674 

n  ' 

44 

433226 

47 

983416 

.59 

449810 

8 

06 

550190 

.6 

45 

433675 

46 

983381 

-59 

450294 

8 

06 

549706 

i5  , 

46 

434122 

45 

983345 

-59 

4^0777 

8 

o5 

549223 
548740 

14 

47 

434569 

44 

983309 

.59 

451260 

8 

04 

.3 

48 

435016 

44 

983273 

.60 

451743 

8 

o3 

548207 

12 

^9 

435462 

43 

983238 

-60 

452220 

8 

02 

547775 

II 

5o 

435908 

42 

983202 

-60 

452706 

8 

02 

547294 

.0 

5i 

9-436353 

41 

9-983166  1 

.60 

9-453187 

8 

01 

10-5463.3 

9 

52 

436798 

40 

983 1 3o  1 

-60 

453668 

8 

00 

546332 

8 

i^ 

437242 

40 

9830-94 

.60 

454148 

99 

545302 

I 

54 

437686 

It 

983008 

.60 

454628 

^ 

545372 

55 

438129 

983022 

•  60 

455107 

544893 

5 

56 

438572 

37 

982986 

.60 

455586 

97 

544414 

4 

ll 

439014 

36 

982950 

.60 

456064 

96 

543936 

3 

439456 

36 

982914 

.60 

456542 

96 

543453 

2 

59 

439897 
440338 

35 

982878 

.60 

457019 

95 

542981 

I 

60 

7-34 

982842 

.60 

457496 

7-94 

542004   0  1 

Cosine 

1   D. 

Sino 

Cotang. 

D. 

Tann..   mJ 

(74    DEGREES.) 


'64: 


(10    DEGREES.)      A  TABLE   OF   LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotaiig. 

~1 

o 

9.440338 

7-34 

9-9*^2842 

60 

9-457496 

7-94 

10- 542504 

Ao 

1 

440778 

7 

33 

9S2805 

60 

457973 

7-93 

542027 

2 

441218 

7 

32 

982769 

61 

458449 

7.93 

54.55i 

3 

441658 

7 

3i 

9S2733 

61 

458925 

7-92 

54.075 

57 

4 

442og6 

7 

3i 

9^^2696 

61 

459400 

7-91 

540600 

56 

5 

442535 

7 

3o 

9^2660 

61 

459875 

7.90 

5401 25  i  55 

6 

442973 

7 

29 

982624 

61 

460849 

539651  !  54 

7 

443410 

7 

28 

9S2587 

61 

460823 

7-80  ■ 

530.77  1  53  1 

8 

443847 

7 

27 

982551 

61 

461297 

7-88 

538703  1  52 

9 

4442 ':54 

7 

27 

9S2514 

61 

46.770 

7-88 

538230  !  5i  ! 

10 

444720 

7 

26 

9^2477 

61 

462242 

7-87 

537758  1  5o 

II 

9-445.55 

7 

25 

9-982441 

61 

9-462714 

7.86 

10-537286  i  49 
5368.4  !  48 

12 

445590 

7 

24 

982404 

61 

463 186 

85 

i3 

446025 

7 

23 

982867 

61 

463658 

85 

536342  1  47 

14 

446459 

7 

23 

982331 

61 

464129 

84 

535871  1  46 

i5 

446893 

7 

22 

982294 

61 

464599 

83 

535401  j  45 

i6 

447326 

7 

21 

982257 

61 

465069 

83 

534981  1  44 

n 

447739 

20 

982220 

62 

465539 

82 

534461  !  43 

i8 

448 191 

7 

20 

982183 

62 

466008 

8i 

533992  1  42 

'9 

448623 

7 

;? 

982146 

62 

J66476 

80 

533324  i  41 

20 

449054 

7 

982109 

62 

466945 

80 

533o55  1  40  1 

21 

9-4494S5 

7 

17 

9-982012 

62 

9-4674.3 

]t 

10-532587  1  39 
532120  1  38 

22 

449915 

7 

16 

9S2035 

62 

467880 

23 

45o345 

7 

16 

981998 

62 

468347 

78 

53i653  1  37 

24 

450775 

7 

i5 

981961 

62 

468814 

77 

53 1186  1  36 

25 

451204 

7 

14 

981924 

62 

469280 

76 

530720  ;  35 

26 

45i632 

7 

i3 

981886 

62 

469746 

n 

530254  [  34 

27 

452060 

7 

i3 

98 1 849 

62 

470211 

75 

5297S9  i  33 

28 

452488 

7 

12 

981812 

62 

470676 

H 

529824  1  32 

29 

452qi5 

7 

n 

9^1774 

62 

471 141 

7^ 

528859  j  3i 

3o 

453342 

7 

10 

981737 

62 

471605 

73 

528893 

3o 

3i 

9-453768 

7 

10 

9-981699 

63 

9-472068 

72 

10.527932 

'I 

32 

454194 

7 

09 

981662 

63 

472532 

71 

527468 

33 

454619 

7 

08 

981625 

63 

472995 

71 

527005 

27 

34 

455o44 

7 

07 

981587 

63 

473457 

70 

526543 

26 

35 

455469 

7 

07 

981549 

63 

473919 

69 

526081 

25 

36 

455893 

7 

06 

981512 

63 

474381 

^ 

5256i9 

24 

37 

4563 1 6 

7 

o5 

9-^«474 

63 

474842 

525i58  1  23  1 

3S 

456739 

7 

o4 

981436 

63 

4753o3 

67 

524697 

22 

39 

457162 

7 

04 

98.399 

63 

475763 

67 

524237 

21 

40 

457584 

7 

o3 

981861 

63 

476223 

66 

523777 

20 

41 

9 -458006 

7 

02 

9-981823 

63 

9-476683 

65 

10-523317 

•9 

42 

458427 

7 

01 

981285 

63 

477142 

65 

522858 

18 

43 

458848 

7 

01 

981247 

63 

477601 

64 

522899 

\l 

44 

439268 

7 

00 

98.209 

63 

478059 

63 

521941 

45 

459688 

6 

99 

981171 

63 

478517 

63 

521483 

i5 

46 

460108 

6 

98 

981133 

64 

478975 

62 

521025 

14 

47 

460527 

6 

98 

981095 

64 

479432 

61 

520568 

i3 

48 

460946 

6 

97 

98.057 

64 

479889 

6i 

520111 

12 

49 

461 364 

6 

96 

981019 

64 

480843 

60 

519655 

II 

5o 

461782 

6 

95 

980981 

64 

480801 

59 

519199 

10  1 

5i 

9-462199 

6 

95 

9-980942 

64 

9.481257 

tl 

10-518743 

li 

52 

462616 

6 

94 

980904 

64 

481712 

518288 

53 

463o32 

6 

93 

980866 

64 

482167 

57 

5.7833 

7  1 

54 

463448 

6 

93 

980S27 

64 

482621 

57 

5.7379 

6  ! 

55 

463864 

6 

92 

9807S9 

64 

488075 

56 

5.6925 

5  ; 

56 

464279 

6 

91 

980750  ! 

64 

483529 

55 

516471 

4  ' 

57 

464694 

6 

90 

980712  ; 

64 

488982 

55 

5i6oi8 

3 

58 

465 in8 

6 

r. 

9H0673  i 

64 

484435 

7 

54 

5i5565 

2 

59 

465522 

6 

980635  j 

64 

484887 

53 

5i5ii3  ;  I 

60 

465935 

6-88 

980596 

64 

485339 

7-53 

5i466i  ;  0  1 

1  Cosine 

D. 

Sine 

Cotang-. 

D. 

Tan.e.   M.  | 

(73   DECiREKS.) 


SINES   AND   TANGENTS.       (17    DEGREES.) 


[Mr 

Sine 

D. 

Cosine 

D. 

TMVr. 

D. 

Cotiiiij;.  ! 

r 
o 

9-465935 

6-88 

9-980596 

64 

9-485339 

7-55 

io-5i466i  I  60 

I 

466348 

6-88 

9So5d8 

64 

435791 

7-52 

5.4209  i  5o 
5i3758  1  58 

2 

466761 

6-87 

980519 

65 

486242 

7-5i 

3 

467173 

6-86 

9804^0 

65 

486693 

7-5. 

5.3307  j  57 

4 

467085 

6-85 

980442 

65 

487143 

7-5o 

5.2857 

56 

5 

467996 

6-85 

980403 

65 

487593 

7-49 

5.2407 

55 

6 

468407 

0-84 

980364 

65 

488043 

7-49 

5.1957 

54 

7 

46S817 

6-83 

980325 

65 

488492 

7-48 

5.!5o3 

53 

8 

469227 

6-83 

980286 

65 

488941 

7-47 

5.1009 

52 

9 

469637 

6-82 

980247 

65 

489390 

7-47 

5.06.0 

5i 

10 

470046 

6-81 

980208 

65 

489838 

7-46 

5.0.62 

5o 

II 

9-470455 

6-80 

9-980169 

65 

9-490286 

7-46 

10-5097.4 

49 

12 

470863 

6-8o 

980130 

65 

490733 

7.45 

509267 

48 

i3 

471271 

6-7? 
6-78 

080091 

65 

491 180 

7-44 

508820 

47 

14 

471679 

980052 

65 

491627 

7-44 

5o8373 

46 

i5 

472086 

6-78 

980012 

65 

492073 

7.43 

507927 

45 

i6 

47'M92 

6-77 

979913 

65 

492519 

7-43 

50748. 

44 

;? 

472898 

6-76 

979934 

66 

492965 

7.42 

5o7o35 

43 

473304 

6-76 

979^95 

66 

493410 

7-41 

506090 

42 

'9 

473710 

6-75 

97985D 

66 

493854 

7.40 

5o6.46 

41 

20 

4741 i5 

6-74 

979816 

66 

494299 

7.40 

5o570i 

40 

21 

9-474519 

6-74 

9-979776 

66 

9-494743 

7.40 

io-5o5257 

39 

22 

4749'i3 

6-73 

979737 

66 

495 1 «6 

]:ll 

5048.4 

38 

23 

475327 

6.72 

9796.')8 

66 

495630 

504370 

37 

24 

475730 

6.72 

66 

496073 

7-37 

503927 

36 

25 

476133 

6.71 

919618 

66 

4965 1 5 

7-37 

5o3485 

35 

26 

476536 

"6-70 

979579 

66 

496957 

7.36 

5o3o43 

34 

27 

476938 

6-69 

979539 

66 

497399 

7-36 

5o26o.   33 

28 

477340 

6-69 
6-68 

979499 

66 

497'^4i 

7-35 

502.59   32 
50.7.8  3i 

^9 

477741 

979439 

66 

49'i282 

7-34 

3o 

478142 

6-67 

979420 

66 

498722 

7-34 

50.278  3o 

3i 

9-478542 

6-67 

9-979380 

66 

9-499163 

7-33 

10.500837   29 

32 

4789.12 

6.66 

979340 

66 

499603 

7.33 

5oo397 

28 

33 

4793.i2 

6-65 

979300 

67 

5ooo42 

7.32 

499958 

27 

34 

479741 
480140 

6-65 

979260 

67 

5oo48i 

7.31 

499'^ '9 

26 

3d 

6-64 

979220 

67 

500920 

7-31 

4990*^0 

25 

36 

480539 

6-63 

979.80 

67 

5oi359 

7.30 

498641 

24 

u 

480937 

6.63 

979140 

67 

5o>7o7 

502230 

7-30 

498203 

L'3 

481334 

6-62 

979100 

67 

U 

497765 

22 

39 

48 1 73 1 

6-6i 

979059 

67 

502672 

49-'328  1  21  1 

40 

482128 

6-6i 

979019 

67 

5o3io9 

7-28 

496891 

20 

41 

9-4^2525 

6.60 

9-978979 

67 

9 -503546 

7-27 

10-496454 

:i 

42 

48292 r 

6-59 

978^98 

67 

5039S2 

1-21 

496018 

43 

•  4833 1 6 

6-59 

67 

5o44i8 

7-26 

495582 

17 

44 

4H37I2 

6-58 

978838 

67 

504854 

7-25 

495 1 46 

.6 

45 

4^4107 

6-57 

9788.7 

67 

505289 

7-25 

4947 •>   1 5 

46 

484501 

6-57 

978777 

67 

5o5724 

7-24 

491276   .4 

47 

484895 

6-56 

978736 

67 

5o6i59 

7-24 

493^41   i3 

48 

4^5289 

6-55 

97^96 

68 

506593 

7-23 

493. i)7  '.   12 

49 

4836-i2 

6-55 

978605 

68 

507027 

7.22 

4g:^fr3   . 1 

5o 

486075 

6-54 

978615 

68 

507460 

7.22 

492040  1  .0 

5i 

9- 486^67 

6-53 

9-978574 

68 

'9-507893 

7-21 

10-492107   9 

52 

4S6S60 

6-53 

978533 

68 

5o8326 

7-2. 

491674 

8 

53 

4'<725i 

6-52 

978493 

68 

508759 

7-20 

49 1 2  4 . 

7 

54 

487643 

6.5i 

978452 

68 

509191 

7.19 

490^00 

6 

55 

4H8034 

6-5i 

97«4.i 

68 

509622 

7-19 

490378 

5 

56 

4^8424 

6-50 

97H370 

63 

5ioo54 

T18 

4^9916   4 

u 

488S14 

6-5o 

978329 

68 

510485 

7.. 8 

4H95.5  1  3 

489204 

6-49 

9782 S8 

68 

510916 

7-17 

-  489^84  :  2 

59 

489593 

6-48 

978247 

68 

5n346 

7-. 6 

488654   I 

60 

489982 

6-48 

978206 

68 

5.1776 

7-16 

48S224   0 

Cosine 

D. 

Sine   ! 

D. 

CotuniJT. 

D. 

Tan^.   i  M. 

17 


(72    DEGREES.) 


36 


(18    DEGREES.)      A   TABLE   OF    LOGARITHMIC 


M. 

Sine 

D. 

Cosine  1 

D. 

Tang. 

D. 

Cotang. 

0 

9-489982 

6.48 

9-978206 

68 

9-511776 

7.16 

10-488224 

60 

I 

490371 

6-48 

978165 

68 

512206 

7 

16 

487794  ;  5q 
487365   58 

2 

490709 

6-47 

978 1 24  1 

68 

512635 

7 

i5 

3 

49 1 1 47 

6-46 

978083 

69 

5i3o64 

7 

14 

486936  '  57 

4 

491535 

6-46 

978042 

69 

513493 

7 

14 

486307   56 

5 

491922 

6-45 

978001 

69 

513921 

7 

i3 

4860^9  55 

6 

7 

492308 
492695 
493081 

6-44 
6-44 

977939  • 
977918 

69 
69 

514349 

5«4777 

7 
7 

i3 
12 

485631   54 
485223   53 

8 

6-43 

977877 

69 

5i52o4 

7 

12 

484-96   52 

9 

493466 

6-42 

977835 

69 

51 563 1 

7 

11 

484359   5 1 
483943   5o- 

10 

49385 I 

6-42 

977794 

69 

516057 

7 

10 

II 

9-494236 

6-41 

9-977752 

69 

9.516484 

7 

10 

io-4835i6  1  49 

12 

494621 

6.41 

9777 1 1 

69 

516910 

7 

09 

483090  .  48 

i3 

493005 

6-40 

977669 

^ 

517335 

7 

09 

482665  1  47 

14 

49D388 

6.39 

977628 

69 

517761 

7 

08 

482239  i  46 

i5 

495772 

6-39 
6-38 

9775>s6 

69 

5 1  .^  1 85 

7 

08 

481810   45 

i6 

496154 

977544 

70 

518610 

7 

07 

481390  1  44 

17 

496537 

6.37 

9775o3 

70 

519334 

7 

00 

4^0966 

43 

i8 

496919 

6.37 

977461 

70 

5i94:)8 

7 

06 

480342 

42 

'9 

497301 

6-36 

977419 

70 

519182 

7 

o5 

480118 

41 

20 

4976S2 

6-36 

977377 

70 

52o3:)3 

7 

o5 

479695 

4Q 

21 

9-498064 

6-35 

9-977335 

70 

9.520728 

7 

04 

10-479272 

478849 

39 

22 

498444 

6-34 

977293 

70 

32  1 l5l 

7 

o3 

38 

23 

498825 

6.34 

977231 

70 

521573 

7 

o3 

478427 

h 

24 

499204 

6-33 

977209 

70 

521995 

7 

o3 

478005 

36 

25 

499584 

6-32 

977167 

70 

522417 

7 

02 

477583 

35 

26 

499963 

6-32 

977125 

70 

522^38 

7 

02 

477162 

34 

11 

5oo342 

6.3i 

977083 

70 

523239 

7 

01 

4767.;! 

33 

500721 

6.3i 

977041 

70 

523o8o 

7 

01 

476320 

32 

29 

501099 

6-3o 

976999 

70 

524100 

7 

00 

475900 

3i 

3o 

501476 

6-29 

976937 

70 

524320 

6 

99 

475480 

3o 

3i 

9-5oi854 

6-29 
6.28 

9-976914 

70 

9-324939 

6 

99 

io-475o6i 

29 

32 

50223l 

976872 

523359 
525778 

6 

98 

474641' 

28 

33 

502607 

6.28 

976S3o 

6 

98 

414222  :  27 

34 

502984 

6-27 

976787 

526197 

6 

97 

473^;o3   26 

35 

5o336o 

6-26 

976743 

526613 

6 

97 

473385   25 

36 

5o3735 

6-26 

976702 

527333 

6 

96 

472967   24 

37 

5o4 1 1 0 

6-25 

976660 

527431 

6 

96 

472349  i  23 

38 

504485 

6-25 

976617 

527868 

6 

95 

472132  i  22 

39 

504860 

6-24 

976374 

528285 

6 

95 

471715  1  21 

40 

5o5234 

6-23 

976532 

528702 

6 

94 

471298   20 

41 

9 -503608 

6-23 

9-976489  . 

9-529119 

6 

93 

10-470881  1  19 
470465   18 

42 

505981 

6-22 

976446 

529335 

6 

93 

43 

5o6354 

6-22 

976404 

529900 

6 

93 

470o5a   17 

44 

506727 

6-21 

976361 

53o366 

6 

92 

469634 

16 

45 

507099 

6-23 

676318 

530781 

6 

9« 

469219 

ID 

46 

507471 

6-20 

976275 

531196 

6 

91 

46SS04 

14 

% 

507S43 

6- 19 

976232 

531611 

6 

90 

46H3S9 

i3 

5o82i4 

6-19 

976189 

532025 

6 

g 

467973 

12 

49 

5o8585 

6.18 

976146 

532439 

6 

467361 

II 

5o 

508956 

6-i8 

976103 

532853 

6 

89 

467147 

10 

5i 

9.509326 

6.17 

9-976060 

9-533266 

6 

88 

10-466-34 

t 

52 

509696 

6-16 

976017 

533679 

6 

88 

466321 

53 

5 10065 

6-16 

973974 

534092 

6 

87 

463908 

7 

54 

510434 

6.i5 

973930  ; 

534304 

6 

87 

463496 
463084 

6 

55 

5io8o3 

6-i5 

973887 

534916 

6 

86 

5 

56 

511172 

6-14 

975844 

533323 

6 

86 

464672 

4 

57 

5ii54o 

6-i3 

973800 

535739 

6 

85 

■464261 

3 

58 

511907 

6-i3 

973757  j 

536 1 3o 

6 

85 

463830 

2 

59 

512275 

6.12 

973714  1 

536561 

6 

84 

463430 
463028 

I 

60 

512642 

6-12 

973670 

536972 

6-84 

0 

Cosine 

D. 

Sine 

D. 

Cotang. 

'   D. 

Tang. 

M. 

(71  DEGREES.) 


SINES   AXD   TANGENTS.      (19    DEGREES.) 


37 


JM. 

Sine 

D. 

Cosiuo 

D. 

Tang. 

1>. 

Cotuug.  , 

0 

9-512642 

6-12 

9.975670 

73 

9-536972 

6-84 

10-463028 

60 

I 

5 1 3009 

6 

II 

975627 

73 

537382 

6 

83 

4626.8 

58 

2 

5.3375 

6 

II 

975583 

"7^ 

537792 

6 

83 

4622J8 

3 

5i374i 

6 

10 

973339 

•73 

538202 

6 

82 

461198 

57 

4 

514107 

6 

09 

975496 

•^^ 

5386.1 

6 

82 

46. 3 -(9 

56 

5 

514472 

6 

°2 

9734-32 

73 

539020 

6 

81 

460980 

55 

6 

514837 

6 

o§ 

975408 

73 

539429 

6 

81 

46037. 

54 

7 

5l5202 

6 

08 

973365 

73 

539837 

6 

80 

460.63 

53 

8 

5 1 5566 

6 

07 

•  975321 

73 

543245 

6 

80 

459733 

52 

9 

5 1 5930 

6 

07 

973277 

.73 

540653 

6 

79 

4593  47 

5. 

10 

516294 

6 

06 

975233 

73 

541061 

6 

79 

438939 

5o 

II 

9-5i6657 

6 

o5 

9.975189 

73 

9-541468 

6 

78 

io-45S5i2 

49 

12 

517020 

6 

o5 

973.45 

H 

541875 

6 

78 

438.23 

48 

i3 

517382 

6 

04 

973101 

73 

54228. 

6 

77 

457719 

47 

14 

517745 

6 

04 

973057 

73 

542688 

6 

77 

457812 

46 

i5 

518107 

6 

o3 

9730.3  1 

73 

543094 

6 

76 

456906 

45 

i6 

518463 

6 

o3 

974969 

74 

543499 

6 

76 

456301 

44 

'7 

518829 

6 

02 

974923 

74 

543905 
5443.0 

6 

75 

436093 

43 

i8 

5! 9  loo 

6 

01 

974880 

74 

6 

75 

455690 

42 

»9 

5I955I 

6 

01 

974836  I 

74 

544715 

6 

74 

453285 

41 

20 

5I99II 

6 

00 

974792  j 

74 

545.19 

6 

74 

454881 

40 

21 

9-520271 

6 

00 

9-974748  1 

74 

9-545524 

6 

73 

10-454476 

It 

22 

52o63i 

5 

99 

974703  1 

74 

545928 

6 

73 

454072 

23 

520990 

5 

99 

974639  1 

74 

546331 

6 

72 

453669 

37 

24 

521349 

5 

98 

974614 

74 

546735 

6 

72 

453265 

36 

25 

521707 

5 

98 

974570 

74 

547138 

6 

71 

452862 

35 

26 

522066 

5 

97 

974525 

74 

547540 

6 

7' 

432460 

34 

27 

522424 

5 

96 

974481 

74 

547943 

6 

70 

432037 

33 

28 

522781 

5 

96 

974436 

74 

548343 

6 

70 

45.655 

32 

29 

523 i 38 

5 

9D 

974391 

74 

548747 

6 

69 

451253 

3i 

3o 

523495 

5 

95 

974347  i 

75 

549149 

6 

69 

45o85i 

3o 

3i 

9-523852 

5 

94 

9-974302 

75 

9-549550 

6 

68 

io-45o45o 

29 

32 

524208 

5 

94 

974257 

75 

549951 

6 

68 

450049 

28 

33 

524564 

5 

93 

9742.2  ! 

75 

53o332 

6 

67 

449')48 

27 

34 

524920 

5 

93 

974167  1 

75 

550732 

6 

67 

449 -'48 

26 

35 

520275 

5 

92 

974122  j 

75 

55.132 

6 

66 

44-<rf48 

25 

36 

52  5630 

5 

91 

974077 

75 

55.532 

6 

66 

448448 

24 

37 

525984 

5 

91 

974032 

75 

55.932 

6 

65 

4i8o48 

23 

38 

526339 

5 

90 

973987 

7J 

53235. 

6 

65 

447649 

22 

39 

526693 

5 

?9 

973942 

73 

552750 

6 

65 

4472  30 

2. 

40 

527046 

5 

973897 

73 

553.49 

6 

64 

44683. 

20 

41 

9-527400 

5 

^9 

9-973852 

75 

9-553343 

6 

64 

10-446432 

'9 

42 

527753 

5 

88 

973807 

75 

553946 

6 

63 

446034 

18 

43 

528105 

5 

88 

973761 

75 

554344 

6 

63 

445656 

17 

44 

528458 

5 

87 

973716 

76 

554741 

6 

62 

445259 

16 

45 

528810 

5 

87 

973671  1 

76 

555.39 

6 

62 

444861 

i5 

46 

529161 

5 

86 

973623  i 

76 

555336 

6 

61 

444464 

14 

47 

529313 

5 

86 

973380  ' 

l(^ 

553933 
556329 

6 

61 

444067 

i3 

4^ 

529864 

5 

85 

973535  ; 

76 

6 

60 

44367. 

12 

49 

53021 5 

5 

85 

973489  i 

76 

556725 

6 

60 

443273 

.1 

DO 

53o565 

5 

84 

973444 

76 

557.21 

6 

59 

442879 

10 

5i 

9 -5309 1 5 

5 

84 

9-973398  1 

76 

9-557317 

6 

59 

10-442483 

t 

52 

531265 

5 

83 

973332  i 

76 

5579.3 

6 

59 

442087 

53 

53i6i4 

5 

82 

973307  ; 

76 

5583o8 

6 

5S 

441692 

7 

54 

531963 

5 

82 

97326.  1 

76 

553702 

6 

58 

441298 

6 

55 

532312 

5 

81 

9732.5  ; 

76 

559097 

6 

57 

440903 

5 

56 

532661 

5 

8i 

-973.69  ; 

76 

55949. 

6 

57 

440309 

4 

^7 

533009 

5 

80 

973124  1 

76 

559885 

6 

56 

4401  I  3 

J 

58 

533357 

5 

80 

973078  1 

76 

560279 

6 

56 

439721 

2 

59 

533704 

5 

79 

973o32  [ 

77 

560673 

6 

55 

439327 

I 

1  60 

534052 

5.78 

972986  1 

77 

56 1 066 

6  55 

438934 

Tan-. 

0 
M. 

Cosine 

U 

Sine   ! 

D. 

Cotancr. 

D. 

(70    DEGREES.) 


^H 


(20   DEGREES.)     A  TABLE   OF   LOGAR1THM:IO 


0 

Si.ic- 
9 -534052 

.1). 

Cosine 

D. 

T.U1.;. 

D. 

Cut....  I 

5.78 

9-972986 

•77 

9.561066 

6.55 

10438934  60 

I 

5343Q9 

5 

77 

97^940 

•77 

56.459 

6 

54 

43S341   59 

2 

534745 

5 

77 

972^94 

•77 

56.»Di 

6 

34 

438.49  58 

3 

535092 

5 

77 

972S48 

•77 

562244 

6 

53 

437750 

57 

4 

53543^ 

5 

76 

972802 

•77 

562636 

6 

53 

437864 

56 

5 

5i5-iS3 

5 

76 

972755 

•77 

563028 

6 

53 

4369-2 

55 

6 

536129 

5 

75 

972709 

•77 

563419 

6 

52 

43638. 

54 

7 

536474 

5 

74 

972663 

•77 

5638.1 

6 

52 

486.89 

53 

8 

5368 18 

5 

74 

972617 

•77 

564202 

6 

5i 

43579^ 

52 

g 

537163 

5 

73 

972570 

•77 

564592 

6 

5. 

435408 

5i 

10 

537507 

5 

73 

972524 

•77 

5649^3 

6 

5o 

4350.7 

5o 

II 

9.537851 

5 

72 

9.972478 

•77 

9.565373 

6 

5o 

10.434627 

S 

12 

535194 

5 

72 

972481 

•1^ 

565763 

6 

49 

484287 

i3 

53353S 

5 

71 

972J85 

.78 

566153 

6 

49 

433b47 

47 

14 

538S80 

5 

71 

972338 

•7? 

566542 

6 

49 

433458 

46 

i5 

539223 

5 

70 

972291 

•"^2 

566982 

6 

48 

433068 

45 

i6 

539565 

5 

70 

972240 

•"^^ 

567820 

6 

48 

482680 

44 

17 

539907 

5 

69 

972198 

•''« 

567709 

6 

47 

432291 

43 

18 

540249 

5 

69 

97201 

•78 

56809^ 

6 

47 

43.902 

42 

19 

540*90 

5 

68 

972105 

.78 

568486 

6 

46 

43.0.4 

41 

20 

540931 

5 

68 

972058 

.78 

568873 

6 

46 

43.127 

40 

2. 

9-541272 

5 

67 

9-072011 

•78 

9-569261 

6 

45 

10.480739 

It 

22 

54i6i3 

5 

67 

971964 

•  78 

569648 

6 

43 

43o302 

23 

541953 

5 

66 

97'9«7 

•■^s 

570035 

6 

45 

429965 

11 

24 

542293 

5 

66 

971 870 

•"'^ 

570422 

6 

44 

429578 

25 

542632 

5 

65 

97i«23 

'H 

570809 

6 

44 

429.91 

35 

26 

542971 

5 

65 

971776 

•78 

571193 

6 

43 

428805 

34 

27 

5433 10 

5 

64 

971729 

•79 

57i5di 

6 

43 

423419  i  ii   1 

28 

543649 

5 

64 

9716S2 

•79 

57.967 

6 

42 

428033  i  32  1 

29 

543987 

5 

63 

971635 

•79 

572352 

6 

42 

427648 

■J-  I 

3o 

544325 

5 

63 

971588 

•79 

572738 

6 

42 

427262 

3d  1 

3i 

9-544663 

5 

62 

9.971540 

•79 

9.573.28 

6 

41 

10.426877 

^9  1 

32 

543000 

5 

62 

97 > 493 

•79 

573507 

6 

41 

426493  28 

33 

545338 

5 

61 

97 '446 

•79 

578892 

6 

40 

426108  27 

34 

543674 

5 

61 

971398 

•79 

574276 

6 

40 

425724  26 

35 

54601 1 

5 

60 

97'35i 

•79 

574060 

6 

39 

420340  t  25 

36 

546347 

5 

60 

97i3o3 

•79 

575044 

6 

39 

424936  24 

ill 

546683 

5 

59 

971256 

•79 

573427 

6 

39 

424373   23 

547019 

5 

59 

971208 

•79 

5738.0 

6 

38 

424190 

22 

39 

547354 

5 

58 

971 161 

•79 

576.93 

6 

38 

423 io7 

21 

40 

547689 

5 

53 

97iii3 

•79 

576576 

6 

37 

423424 

20 

41 

9.548024 

5 

57 

9.971066 

.80 

9-576958 

6 

37 

10.42304. 

^2 

42 

54S359 

5 

57 

971018 

.80 

577341 

6 

36 

422659 

18 

43 

548693 

5 

56 

970970 

.80 

577723 

6 

36 

422277 

n 

44 

549027 

5 

56 

970922 

.80 

578104 

6 

36 

42 . 896 

.6 

45 

549360 

5 

55 

970674  1 

.80 

575^486 

6 

35 

42.0.4 

.5 

46 

549693 

5 

55 

970827 

.80 

578867 

6 

35 

421.33 

14 

% 

550026 

5 

54 

970779 

•80 

579248 

6 

34 

420702 

.3 

55o359 

5 

54 

970781 

.80 

579629 

6 

34 

420871 

12 

49 

550692 

5 

53 

970683 

•80 

580009 

6 

34 

419991 

II 

56 

551024 

5 

53 

970635 

•  So 

580889 

6 

33 

41961. 

10 

5i 

9.55i356 

5 

52 

9-970536 

.80 

9.580769 

6 

33 

10-41923. 

t 

52 

55.687 

5 

52 

970538  j 

.80 

581.49 

6 

32 

4 1 885 1 

53 

552018 

5 

52 

970490  1 

.80 

58.528 

6 

32 

418472 

7 

34 

552349 

5 

5i 

970442  1 

.80 

581907 

6 

32 

418093 

6 

55 

552680 

5 

5i 

970894  1 

.80 

582286 

6 

3i 

4i77'4 

5 

56 

553oio 

5 

5o 

970845  1 

•  81 

582665 

6 

3i 

4.7335 

4 

U 

553341 

5 

5o 

970297  ; 

.81 

583043 

6 

3o 

416957 

553670 

5 

49 

970249  i 

.81 

583422 

6 

3o 

4.6378 

59 

554000 

5 

49 

970200  1 

•81 

583800 

6 

29 

416200 

1 

6c 

1 

554329 

5-48 

970102 

.81 

584177 

6-29 

413823 

1 

Cos.ne 

D. 

Sine 

D. 

<:o:iv.vy. 

D. 

Tan^.  :  L£.  ! 

(69   DEGREES.) 


SIXES   AXD   TAXGEXTS.      (21    DEGREES.; 


39 


"m. 

Sine 

D. 

Cosine 

D. 

Taivr. 

D. 

Cot.uii^. 

60 

0 

9-554329 

5.48 

9.970152  1 

81 

9-5^4177 

6.29 

10.4.5823 

I 

554658 

5 

48 

970103 

81 

584555 

6 

11 

4.3445 

59 

2 

554987 

5 

47 

970055  j 

81 

584932 

6 

4i5o63 

5? 

3 

55531 5 

5 

47 

970006  j 

81 

5S5309 

6 

28 

414691 

57 

4 

555643 

5 

46 

969957 

81 

585686 

6 

27 

4143.4 

56 

5 

555971 

5 

46 

969909  I 

81 

586062 

6 

27 

4.8988 

55 

t 

556299 

5 

45 

969860 

81 

586439 

6 

27 

4i3i6i 

54 

7 

556620 

5 

45 

96981 1  I 

81 

5868 1 5 

6 

26 

4.3i85 

53 

8 

556953 

5 

44 

969762  { 

8i 

587190 

6 

26 

412^.0 

32 

9 

5572^0 

5 

44 

969714  1 

81 

587566 

6 

25 

4124U 

5i 

lO 

557606 

5 

43 

969665 

81 

587941 

6 

25 

412039 

5o 

II 

9.557932 

5 

43 

9.969616 

82 

9 -5,383 1 6 

6 

25 

10.41.684 

49 

12 

558258 

5 

43 

969567  1 

82 

588691 

6 

24 

41 .809 

d 

i3 

558583 

5 

42 

969518 

82 

589066 

6 

24 

4.0934 

47' 

14 

558909 

5 

42 

969469 

82 

589440 

6 

23 

4 1 O36o 

46 

i5 

559234 

5 

41 

969420  j 

82 

5898.4 

6 

23 

4.0186 

45 

i6 

559558 

5 

41 

969370 

82 

590188 

6 

23 

409812 

44 

n 

559S83 

5 

40 

967321 

82 

590562 

6 

22 

409438 

43 

i8 

560207 

5 

40 

969272 

82 

590935 

6 

22 

400065 

42 

'9 

56o53i 

5 

39 

969223 

82 

591808 

6 

22 

4o«692- 

41 

20 

56o855 

5 

39 

969173 

82 

59 I 68 I 

6 

21 

4083.9 

40 

21 

9.561178 

5 

38 

9.969124  1 

82 

9-592054 

6 

21 

10-407946 

ll 

22 

56i5oi 

5 

38 

969075  1 

82 

592426 

6 

20 

407374 

2j 

561824 

5 

37 

969025  1 

82 

592798 

6 

20 

407202 

ll 

24 

562146 

5 

37 

968976  I 

82 

593.70 

6 

'9 

406  ■<29 

20 

562468 

5 

36 

968926  1 

83 

593542 

6 

18 

406458 

35 

26 

562790 

5 

36 

968877  ! 
968827  1 

83 

5939.4 

6 

406086 

34 

27 

563II2 

5 

36 

83 

594285 

6 

18 

4057 1 5 

33 

2:^ 

•  563433 

5 

35 

96877-7 
968728 

83 

594656 

6 

.8 

405344 

32 

29 

563-55 

5 

35 

83 

595027 

6 

17 

4049-^3 

3i 

So 

554075 

5 

34 

968678 

83 

595398 

6 

n 

404602 

3o 

3i 

9.564396 

5 

34 

9.968628 

83 

9-595768 

6 

17 

io.4oi232 

29 

32 

5647" 6 

5 

33 

968578 

83 

596188 

6 

i6 

4o3-(62 

2§ 

33 

565o36 

5 

33 

96^528 

83 

596508 

6 

16 

408492 

27 

34 

56)356 

5 

32 

968479 

83 

596878 

6 

16 

4o3.22 

26 

33 

565676 

5 

32 

968429 

83 

597247 

6 

i5 

402753 

25 

36 

565995 

5 

3i 

968879 

83 

597616 

6 

i5 

4023 -!4 

24 

ll 

566314 

5 

3i 

968329  1 

83 

597985 

6 

i5 

40201 5 

23 

38 

566632 

5 

3i 

968278 

Si 

59S354 

6 

14 

4oi6<6 

22 

39 

566951 

5 

3o 

968228 

84 

598722 

6 

14 

401278 

21 

40 

567269 

5 

3o 

968178 

84 

599091 

6 

13 

400909 

20 

41 

9.567587 

5 

29 

9.968123 

84 

9-599459 

6 

.3 

10.400541 

'9 

42 

567904 

5 

29 

968078 

84 

399827 

6 

i3 

400.73 

18 

43 

5682  2  2 

5 

28 

968027 

84 

600194 

6 

12 

399806 

17 

44 

568539 

5 

28 

9''7977 

84 

6oo562 

6 

12 

399  i33 

16 

45 

568856 

5 

28 

967927 

84 

600929 

6 

II 

39907 1 

i5 

46 

569172 

5 

27 

967^76 

84 

601296 

6 

1 1 

39-^704 

14 

47 

569488 

5 

27 

967826  1 

84 

601662 

6 

1 1 

398338 

i3 

48 

569804 

5 

26 

9^^7775  i 

84 

602029 

6 

10 

39-9^1 

12 

^9 

570120 

5 

26 

967720  [ 

84 

602895 

6 

10 

3976.5 

II 

Do 

570435 

5 

25 

967674 

84 

602761 

6 

10 

397289 

10 

5i 

9.570751 

5 

23 

9-967624 

84 

9 -603 127 

6 

09 

10-396873 

t 

02 

53 

57.066 

5 

24 

967573  1 

84 

603493 

6 

09 

396507 

57i38o 

5 

24 

967522  1 

85 

6o3858 

6 

09 

396142 

7 

54 
55 

571695 

5 

23 

967471 

85 

604223 

6 

08 

393777 

6 

572009 

5 

23 

967421 

85 

604588 

6 

08 

3954.2 

5 

56 

572323 

5 

23 

967370  ! 

85 

604953 

6 

07 

395047 

4 

ll 

572636 

5 

22 

967319  : 

85 

6o53i7 

6 

07 

394683 

3 

572950 

5 

22 

967268  1 

85 

6o5682 

6 

07 

39^3. 8 

2 

59 

573263 

5 

21 

967217  j 

85 

606046 

6 

06 

398954 

I 

60 

573575 

5.21 

967166  j 

85 

606410 

6.06 

393590 

0 

Cosine 

D. 

Sine   ■ 

D. 

Cotaij'jr. 

D. 

Tan-. 

M. 

(68   DKGKEES.) 


40 


(22   DEGREES.)     A   TABLE    OF   LOGARITHMIC 


M. 

1   Sine 

D. 

j  Cosine  1 

D. 

Tanor. 

D. 

Cotang. 

60 

0 

j  9073575 

5.2, 

9-967166 

.85 

9-606410 

6.06 

10-393590 

I 

1   573^88 

5.20 

967115 

•  85 

,   606773 

6-06 

393227  1  59 
392S63  ;  58 

2 

i   574200 

5-20 

967064 

.85 

607137 

6-05 

3 

1   5745.2 

5. .9 

9670.3 

•  85 

607500 

6^o5 

392  "ioO  j  57 

4 

i   •^74S24 

5-'9 

966961 

•  85 

607S63 

6-04 

392.37  1  56 

5 

5751 36 

5-19 

9669.0 

•  85 

60^225 

6-04 

39.775 

55 

6 

5734-i7 

5-18 

966859 
966808 

•85 

6o8583 

6  04 

391412 

54 

7 

575758 

5-18 

•  85 

608950 
6093 1 2 

6-o3 

39.050  '  53  1 

8 

1   576069 

5-, 7 

966756 

•  86 

6-o3 

390638   52 

c. 

576379 

5-17 

966105 

-86 

609674 

6-o3 

390326  '  5i 

10 

576689 

5-i6 

966653 

•  86 

6 1 oo36 

6-02 

389964 

5o 

II 

9-576999 

5-16 

9-966602 

•  86 

9-610397 

6-02 

io-3«96o3 

t 

12 

377309 

5.16 

966550 

•86 

6.0759 

6-02 

389241 

i3 

577618 

5-i5 

966499 

•  86 

611120 

6-01 

3SS880 

47 

14 

577927 

5. .5 

9^>M47 

•  86 

6114H0 

6-01 

388520 

46 

i5 

578236 

5. .4 

966395 

•  86 

6..841 

6-01 

388,59 

45 

i6 

578545 

5. ,4 

966344 

•  86 

6.2201 

6-00 

387799 

44 

'7 

578853 

5-,3 

966292 

.86 

6.256. 

6- 00 

387439 

43 

i8 

579162 

5-13 

966240 

•  86 

6.2021 

6-00 

38-079 

42 

19 

5^9470 

5-13 

966188 

•  86 

6.32S. 

5-99 

3R6-.9 

41 

20 

579777 

5-12 

966136 

•  86 

6.3641 

5-99 

386339 

40 

21 

9-5%o85 

5-12 

9-966085 

•87 

9-614000 

5-98 

10-386000 

It 

22 

580392 

5-II 

966033 

•87 

6,4359 

5.98 

385641 

23 

580699 

5. II 

965981 

•87 

6,4718 

5-98 

385282 

37 

24 

58rooo 

5. II 

965928 
965876 

•87 

6i5o77 

5-97 

3«4923 

35 

25 

58.3.2 

5.10 

•87 

6.5435 

5-97 

384565 

35 

26 

58.6.8 

5.10 

965824 

•87 

6.5^q3 

5-97 

384207 

34 

li 

58.924 

5.09 

965772 

•87 

6.6i5i 

5-96 

3'^3Sig 

33 

582229 

5.09 

965720 

•87 

6i65o9 

5.96 

383491 

32 

29 

582^35 

Vol 

965668 

.87 

6.6867 

5-96 

383.33 

3i 

3o 

582840 

9656 1 5 

•87 

6.7224 

5-^5 

3S2776 

3o 

3i 

9-583.45 

5.08 

9-965563 

•87 

9-6,75^^2 

5-95 

io-3«24i8 

11 

32 

5'^3449 

5.07 

965511 

.87 

6,7939 

5-95 

3^2061 

33 

5^3754 

5.07 

965458 

•87 

6.S295 

5-94 

3«,7o5 

27 

34 

584058 

5.06 

965406 

•87 

6,8652 

5-94 

38,348 

26 

35 

584361 

5.06 

965353 

•  88 

6,9008 

5-94 

380992 

25 

36 

584665 

5-06 

965301 

•  88 

6,9364 

5.93 

38o636 

24 

37 

584968 

5-o5 

965248 

•  88 

61972, 

5-93 

3802-9 

23 

38 

58-1272 

5-o5 

965.95 

•  88 

620076 

5-93 

379924 
379568 

22 

39 

585574 

5-04 

965.43 

.88 

620432 

5-92 

21 

40 

585377 

5-04 

965090 

•  88 

620787 

5.^2 

3792.3 

20 

41 

9-586.79 

5-o3 

9-965037 

•  88 

9-621142 

5-92 

10-378858 

]t 

42 

586 ',82 

5-o3 

9^349^4 

•  88 

621497 

5.91 

3785o3 

43 

586783 

5-03 

964931 

.88 

621852 

5.91 

378148 

17 

44 

58io85 

5-02 

964879 

•  83 

622207 

5-90 

377793 

16 

45 

5873% 

5 -02 

964826 

•  88 

622561 

5-90 

377439 

i5 

46 

58i688 

5-0. 

964773 

•  88 

622915 

5-90 

377085 

14 

il 

587989 

5-01 

964719 

•  88 

623269 
623623 

5-89 

376731 

i3 

58«2^.9 

5-01 

964666 

•  89 

5-f>9 

376377 

12 

49 

588590 

5-00 

9V.613 

•  89 

623976 

376024 

II 

5c 

588890 

5-00 

964560 

•  89 

624330 

375670 

10 

5i 

9-589190 

4-99 

9-964507 

.89 

9-624683 

5-88 

10-3753.7 

t 

52 

5^9^89 

4-99 

964454 

•  89 

625o36 

5-88 

374964  1 

53 

5S97.S9 

t^ 

964400 

.89 

625388 

5-87 

374612  j  7 

54 

590088 

9'^4347 

.89 

625741 

5-87 

374259  I  6 

55 

590387 

4-9^ 

964294 

•  89 

626093 

5-87 

373907   5 

56 

590686 

4-97 

964240 

•  89 

626445 

5-86 

373555   4 

57 

590984 

4-97 

964187 

.89 

626797 

5-86 

3^3203   3 

58 

59.2S2  1 

4-97 

964.33 

•  89 

627149 

5-86 

3^2851 

2 

^ 

59.580  ' 

4-96 

964080 

.89 

62750T 

5-85 

3^2499 

I 

60 

591878 
Cosine 

4.96 

964026 

.89 

627Fr52 

5-85 

372148   0 

D. 

Sine 

D. 

rot.uiir. 

D. 

Tan?.  !  M. 

(67    DEGREES.) 


SINES  AND   TANGENTS.      (23    DEGREES.) 


41 


M. 

Sine 

D.   1 

Co.sine  ! 

D. 

Tiin<,'.   1 

D. 

Cotunr;. 

1 

1 

0 

9.591878 

4-96  ' 

9-964026 

.89 

9-627852 

5-85 

10-372148  1 

771 

I 

592176 

4-95  1 

963972 

•89  ! 

62S203 

5-85  , 

371797  !  59   ! 

2  < 

592473 

4-95 

963919 

.89  1 

628554  1 

5-85 

371446  :  58 

3  i 

592; 70 

4-95 

963865 

.90  ! 

■  628905  i 

5-84 

371095  1  57 

4  i 

593067  1 

4-94 

96381 1 

.90  ; 

6292 j5  I 

5-84 

370745 

56 

5 

593363 

4-94 

963757 

.90  j 

629606 

5-83  ! 

370394 

55 

6 

59360^ 

4-93 

963704 

.90  j 

629956 

5-83 

370044 

54 

7  1 

5939)3 

4-93 

963650 

.90 

63o3o6 

5-83 

369694 

53 

«  ; 

594251  1 

4-93 

963596 

.90  1 

63o656 

5-83 

369344   52  1 

9  i 

594547  ! 

4-92 

963542 

.90  j 

63ioo5 

5-82 

^'^^]       ' 

DI 

lO 

594842 

4-9^ 

963488 

.90  I 

63i355 

5.82 

368645 

5o 

11 

9-595i37 

4-91 

9-963434 

.90 

9-631704 

5-82 

10-368296 

49 

12 

595432  ; 

4-91 

963379 

.90  j 

633053 

5-8i 

367947 

48 

13 

595727 

4-91 
4-90 

96332^ 

.90 

632401 

5-8i 

367099 

^I 

U 

596021 

963271 

.90 

6'3275o 

5-8i 

367200 

46 

i5 

59631 5 

4-90 

968217 
963 1 63 

.90 

633098 

5-8o 

366902 

45 

i6 

596609 

4-89 

.90 

633447 

5-8o 

366553 

44 

17 

59'59o3 

4-89 

963108 

.91 

633795 

5-8o 

3662o5 

43 

i« 

597196 

4-89 

963 0 54 

.91 

634143 

5.79 

365857 

42 

'9 

597490 

4-88 

962999 

.91 

634490 

5-79 

360DIO 

41 

20 

597783 

4-88 

962945 

■91 

634838 

5-79 

365 162 

40 

21 

9-598075 

4-87 

9-962890 

•91 

9-635i85 

5.78 

10.364815 

ll 

22 

59^368 

4-87 

962886 

.91 

635532 

5-78 

364468 

23 

598660 

4-87 

962781 

.91 

635879 

5-78 

364I2I 

37 

24 

597952 

4-86 

962727 

•91 

636226 

5-77 

363774 

36 

25 

599244 

4-86 

962672 

.91 

636572 

5-77 

363428 

35 

26 

599536 

4-85 

962617 

.91 

636919 

5-77 

363o8i 

34 

27 

599.27 

4-85 

962562 

.91 

637265 

5-77 

362735 

33 

28 

600118 

4-85 

962508 

.91 

637611 

5.76 

362389 

32 

29 

600409 

4-84 

962453 

.91 

637956 

5-76 

362044 

3i 

3o 

600700 

4-84 

962398 

.92 

638302 

5-76 

361698 

30 

3i 

9-600990 

4-84 

9-962343 

.92 

9-638647 

5.75 

io-36i353 

29 

32 

6012  30 

4-83 

962288 

.92 

638992 

5-75 

361008 

28 

33 

601570 

4-83 

962233 

.92 

689337 

5-75 

36o663 

27 

34 

601860 

4-82 

962178 

.92 

639682 

5-74 

36o3i8 

26 

35 

6j2i5o 

4-82 

962123 

.92 

640027 

5-74 

359973 

25 

36 

602439 

4-82 

962067 

.92 

640371 

5.74 

359629 

24 

^ 

602728 

4-8i 

962012 

.92 

6407 1 6 

5-73 

359284 
358940 
358096 

23 

38 

6o3oi7 

4-8i 

961907 

.92 

641060 

5-73 

22 

39 

6o33o5 

4-81 

961902 

.92 

641404 

5.73 

21 

40 

603594 

4-80 

96 1 846 

•92 

641747 

5-72 

358253 

20 

41 

9-603382 

4-8o 

9-961791 

•  92 

9-642091 

5-72 

10-357009 

»9 

42 

604170 

4-79 

961735 

.92 

642434 

5-72 

357566 

18 

43 

604457 

4-79 

961680 

■92 

642777 

5-72 

357223 

n 

44 

604745  ■ 

4-79 

961624 

•93 

643120 

5-71 

356880 

16 

45 

6o5o32 

4-78 

961560 
96i5i3 

.93 

643463 

5-71 

356537 

i5 

46 

6o53i9 

4-78 

•93 

648806 

5-71 

35619', 

1 4 

47 

6o56o6 

4-78 

961458 

.93 

6441 i8 

5-70 

355852 

i3 

48 

60)892 

4-77 

961402 

•93 

644490 

5-70 

355510 

12 

49 

606179 

4-77 

961346 

.93 

644832 

5.70 

355168 

i  II 

5o 

606465 

4-76 

961290 

•93 

645174 

5.69 

354826 

|,0 

5i 

9-606751 

4-76 

9-961235 

-93 

9-6455i6 

5-69 

10.354484 

!  t 

52 

607036 

4-76 

961179 
961123 

-93 

645857 

5-69 

;     354143 

53 

607322 

4-75 

.93 

646199 

5-69 

353301  1  7  1 

54 

607607 

4-75 

961067 

.93 

646540 

5-68 

353460 

6 

55 

607892 

4-74 

96 1 0 1 1 

-93 

646881 

5-68 

353119 

5 

56 

608177 

4-74 

960955 

.93 

1   647222 

5-68 

!    352778 

4 

u 

608461 

4-74 

960899 

'  .93 

1   647562 

5-67 

352438 

3 

608745 

4-73 

960843 

i  -94 

647903 

5-67 

i   3)2097 

2 

59 

609029 
609313 

4-73 

960786 

1  -94 

648243 

5-67 

'   35.7)7 

I 

60 

4-73 

960730 

;  .94 

648583 

5-66 

351417 

0 

'  Codine 

!   D. 

Sine 

1  D. 

1  Cota  nir- 

1    D. 

Taiif?.  1  M. 

[QQ    DEGREES.) 


42 


(24    DEGREES.)     A   TABLE    OF   LOGAEITHMIC 


M. 

o 

Sine 

D. 

C.j-iiic 

D. 

Ta.,ir. 

D. 

(.'otiuie. 

1  ■■ 
1 

9-6o93i3 

4-73 

9-960730 

•94 

9-648583 

5-66 

10-3314.7 

I 

609597 

4-72 

960674 

.94 

64S923 

5-66 

331077 

i^ 

2 

609 Soo 

4-72 

960618 

•94 

649263 

5-66 

350737 

3 

610164 

472 

960561 

•94 

649602 

5-66 

350398 

1  ^1 

4 

610447 

4-71 

96o5o5 

•94 

649942 

5-65 

35oo38 

1  56 

5 

610729 

4-71 

960448 

•94 

65o2-ii 

5-65 

349719 

55 

6 

611012 

4-70 

0o392 

.94 

65o62o 

5-65 

349380 

54 

I 

611294 

4-70 

960335 

•94 

6509^9 

5-64 

349041 

53 

611376 

4- 70 

960279 

•94 

65.297 

5-64 

348^03 

1  52 

9 

6ii858 

4-69 

960222 

•94 

65.636 

5-64 

34H364 

I  5i 

10 

612140 

4-69 

960165 

•94 

65.974 

5-63 

348026 

i5o 

II 

9-612421 

469 

9-960109 

.95 

9-652312 

5-63 

10-347688 

\tl 

12 

612702 

4-68 

960052 

.95 

652650 

5-63 

347330 

i3 

612983 

4-68 

939938 

.95 

652988 

5-63 

3470.2 

1  47 

14 

613264 

4-67 

.95 

653326 

5-62 

346')74- 

i  46 

i5 

613545 

4-67 

939882 

.95 

653663 

5.62 

346337 

45 

i6 

6i3825 

4-67 

959825 

.95 

654000 

5.62 

346000 

44 

n 

6i4io5 

4-66 

939768 

.95 

654337 

5.61 

343663 

43 

i8 

614385 

4-66 

9397 1 1 

.95 

654674 

5.61 

345326 

42 

'9 

614665 

4-66 

959654 

.95 

655011 

5.61 

3449^9 

41 

20 

614944 

4-65 

959596 

.95 

655348 

5.61 

34-1652 

40 

21 

9-6i5223 

4-65 

9-959339 

.95 

9-655684 

5.60 

10-344316 

1? 

22 

6i55o2 

4-65 

939482 

.95 

656o2o 

6.60 

3439^^0 

23 

615781 

4-64 

959425 

.95 

656356 

5.60 

343644 

37 

24 

616060 

4-64 

939368 

.95 

656692 

5.59 

343308 

36 

23 

6 16338 

4-64 

939310 

.96 

657028 

5.59 

3429-2 

1  35 

26 

616616 

4-63 

959253 

.96 

657364 

5.59 

343636 

34 

11 

616S94 

4-63 

939195 

.0 

t^, 

5.59 

342301 

33 

6171-2 

4-62 

939.38 

.96 

5-58 

341066 

32 

29 

617430 

4-62 

959081 

.96 

658369 

5-58 

34i63.  :  3i 

3o 

617727 

4-62 

959023 

.96 

658704 

5-58 

34.296 

3o 

3i 

9-618004 

4-6i 

9-958965 

.96 

9-659039 

5-58 

10-340061 

^ 

32 

618281 

4-61 

958908 

.96 

659373 

5.57 

340627 

33 

6 18558 

4-6r 

958850 

•96 

659708 

5.57 

340292 

27 

34 

61 8834 

4-6o 

938792 

.96 

660042 

5.57 

33Q938  1  26 

35 

619110 

4-6o 

958734 

.96 

660376 

5.57 

339624   25 

36 

619386 

4-6o 

93S677 

.96 

660710 

5.56 

339290  i  24 

ll 

619662 

4-59 

958619 

.96 

661043 

5.56 

33^037   23 

6.9938 

4-5? 

958561 

.96 

66.377 

5.56 

33%23   22 

39 

620213 

4-39 

9585o3 

•97 

661710 

6-55 

33^290   21 

40 

620488 

4-55 

958445 

•97 

662043 

5.55 

337937  j  20 

41 

9-620763 

4-58 

9-958387 

•97 

9-662376 

5.55 

10-337624   19 
337291   18 

42 

621038 

4-57 

958329 

•97 

662709 

5.54 

43 

62i3i3 

4.57 

95^211 

•97 

663042 

5.54 

336938  n 

44 

621587 

4-57 

9582! 3 

•97 

663375 

5.54 

336625 

16 

45 

621861 

4-56 

958 1 54 

•97 

663707 

5-54 

336293 

i5 

46 

622135 

4-56 

938096 

•97 

664039 

5-53 

335961 

14 

% 

622409 

4-56 

958o38 

•97 

664371 

5.53 

335629 

i3 

622682 

4-55 

937979 

•97 

664703 

5.53 

335297 

12 

f^ 

622956 

4-55 

957921 
957863 

•97 

665o35 

5-53 

334965   II 

5o 

623229 

4-55 

•97 

665366 

5-52 

334634 

10 

5i 

9-6235o2 

4-54 

9-957804 

■'^ 

9-665697 

5.52 

io.3343o3 

I 

52 

623774 

4-54 

957746 

666029 

5-32 

333971 

53 

624047 

4-54 

937687 

.98 

666360 

5-51 

333640 

7 

54 

624319 

4.53 

937628 

.98 

666691 

5-51  , 

333309 

6 

55 

624591 

4-53 

95-570 

.98 

667021 

5.5i 

3329-9 

5 

56 

624863 

4-53 

9=175.1 

.98 

667352 

5.51  , 

332648 

4 

ll 

625i35 

4-32 

937452 

•9^ 

667682 

5-5o  ; 

3323 18 

3 

625406 

4.52 

95-393 

•gS 

668013 

5.5o  1 

33.987 

3 

P 

-  625677 

4-52 

957335 

.98 

668343 

5-5o 

33.637 

I 

60 

625948 

401 

957276 

.98 

668672 

5-5o 

33.328   0 

1 

Co?:i.e 

D. 

Sine 

D. 

Cotan^r. 

D. 

Ta»^.   M.  i 

(6-5  DEGREES.) 


SINES   AXD   TAXGEXTS.      (25    DEGREES.) 


43 


M. 

0  1 

Sine   • 

D. 

(.'osine 

I). 

Tan-.   i 

I).     ' 

Cotanor.  i 

9-625948  1 

4.51  ! 

9-957276  , 

.98 

9  668673  ' 

5-5o 

io.33i327  60 

I 

626219 

4-31 

9''72i7  ; 

-98 

669002 

5-49 

330998 

It 

2 

626490 

4-5i 

95708 

.98  ; 

669332 

5-49 

33o668 

3 

626760 

4-5o 

9.-)7099 

.98  ' 

669661 

5.4? 

33o339 

57 

4 

627030 

4-5o 

957040 

.98  ; 

669991 

330009   56 

5 

627300 

4-5o 

956981 

.^8  i 

670320 

5.48 

329680   55 

6 

627570 

4-49 

956921 

•99  , 

670649 

5.48 

329351  ' 

54 

7 

621840 

4-49 

956 S62 

•99 

670977 

5.48 

329023 

53 

8 

62S109 

4-49  1 

9568o3 

•99 

671306 

5.47 

32-!69i  ' 

52 

9 

62S378 

4-48 

936744 

•99 

671634 

5.47 

32S366 

5i 

10 

62S647 

4-48 

956684 

•99  • 

671963 

5.47 

328037  : 

5o 

II 

9-628916 

4-47 

9-956625 

•99 

9-672291 

5.47 

10.327709  ' 

49 

12 

629185 

4-47 

956566 

•99 

672619 

5.46 

3273S1  i 

48 

i3 

629453 

4-47 

9565o6 

•99 

672947 

5.46 

327053 

47 

14 

629721 

4-46 

956447 

•99 

673274 

5.46 

326726 

46 

i5 

6299^9 

4-46 

956387 

•99 

673602 

5.46 

326398 

45 

i6 

630257 

4-46 

956327 

•99 

673929 

5.45 

326071 

44 

17 

63o524 

4.46 

956268 

•99 

674257 

5.45 

325743 

43 

18 

630792 

4-45 

956208 

1. 00 

674584 

5-45 

325416 

42 

19 

63 1059 

4-45 

956148 

I -00 

674910 

5-44 

325090 

41 

20 

63i326 

4-45 

956089 

I -00 

675237 

5-44 

324763 

40 

21 

9  -  63 1 593 

4.44 

9-956029 

I  .00 

9-675564 

5.44 

10-324436 

l^ 

22 

63 1 859 

4-44 

955969 

I -00 

675890 

5.44 

324110 

23 

632123 

4-44 

955909 

I -00 

676216 

5.43 

323784 

37 

24 

632392 

4.43 

955849 

I -00 

676543 

5.43 

323457 

36 

25 

632658 

4.43 

955789 

1 .00 

676869 

5.43 

323i3i 

35 

26 

632923 

4-43 

953729 

1.00 

677194 

5-43 

322806 

34 

27 

633189 

4-42 

955669 

1.00 

677520 

5.42 

322480 

33 

28 

633454 

4-42 

955609 
955548 

I.  00 

677846 

5.42 

322154 

32 

29 

633719 

4-42 

1.00 

678171 

5.42 

321829 

3i 

3o 

633984 

4-41 

955488 

1-00 

678496 

5.42 

32i5o4 

3o 

3i 

9-634249 

4-41 

9.955428 

I  -01 

9.678821 

5.41 

10-321179 

11 

32 

634514 

4.40 

955368 

I-OI 

679146 

5.41 

320854 

33 

634778 

4.40 

955307 

I-OI 

679471 

5-41 

32o529 

27 

34 

635042 

4.40 

955247 

I-OI 

679795 

5.41 

320205 

26 

35 

6353o6 

4-39 

955186 

l-OI 

6S0120 

5.40 

319880 

23 

36 

635570 

4-39 

955126 

I-OI 

680444 

5.40 

319556 

24 

37 

635834 

4-3n 
4.3^ 

955o65 

I-OI 

680768 

5.40 

319232 
318908 

23 

38 

636097 

955oo5 

I-OI 

681092 

5.40 

22 

39 

636360 

4-38 

954944 

I-OI 

681416 

5-39 

3 1 8584 

21 

40 

636623 

4-38 

954883 

I-OI 

681740 

5-39 

318260 

20 

41 

9-636886 

4-37 

9.954823 

I-OI 

9.682063 

5.39 

10-317937 

\t 

42 

637148 

4-37 

954762 

I  -01 

682387 

5.35 
5.38 

317613 

43 

6374 r I 

4-37 

954701 

I-OI 

682710 

317290 
316967 

17 

44 

6376-3 

4-37 

954640 

I-OI 

683o33 

5-38 

16 

45 

637935 

4-36 

954579 
954518 

I  -01 

683356 

i  5-38 

316644 

i5 

46 

638197 

4-36 

1-02 

683679 

I  5-38 

3i632i 

i  U 

47 

638458 

4-36 

954457 

1-02 

684001 

!  5-37 

3 1 5999 

:  '3 

48 

638720 

4-35 

954396 

1.02 

684324 

1  5.37 

3 15676 

!  12 

49 

638981 

4.35 

954335 

1.02 

K     684646 

5.37 

3 1 53 54 

i  " 

5o 

639242 

4.35 

934274 

1-02 

684968 

5.37 

3i5o32 

10 

5i 

9 '639503 

4-34 

9.954213 

1-02 

9-685290 

5.36 

10-314710 

I 

52 

639764 

4-34 

954152 

1-02 

685612 

5.36 

314388 

53 

640024 

4-34 

954090 

1-02 

'   685934 

5.36 

314066 

1 

54 

640284 

4.33 

954029 

1-02 

!  6S6255 

5-36 

3 1 3745 

6 

55 

640544 

4-33 

953968 

1   1.02 

j   686577 

5-35 

3i3423 

5 

56 

640804 

4-33 

953906 

1.02 

!   686898 

5.35 

3i3i02 

4 

57 

641064 

4-32 

953845 

1-02 

I   687219 

5-35 

3 1 278 1 

3 

58 

641324 

4-32 

953783 

1-02 

687540 

5.35 

312460 

2 

59 

641584 

4-32 

953722 

1.03 

687861 

5-34 

3i2i39 
311818 

r 

60 

641842 

4-3i 

953660 

l-03 

-  688182 

5.34 

0 

Cosine 

D. 

1   Sine 

1  D. 

1  Cotanc:. 

D. 

i   Tiinir. 

:m. 

♦ 

(6' 

l:  DECK 

?:es.) 

44 


(26   DEGREES.)     A  TABLE   OF   LOGARITHMIC 


M. 

Sine 

'D. 

Cosine   1  D. 

Tang. 

D. 

Cotang. 

1 
60 

0 

9-641842 

4-3i 

9-953660    I 

o3 

9-688182 

5.34 

10-311818 

I 

642101 

4 

3i 

953;io9    i 

o3 

688502 

5 

34 

311498 

ll 

2 

642360 

4 

3i 

953)37    I 

o3 

688823 

5 

34 

311177 

3 

642618 

4 

3o 

953475    I 

o3 

689143 

5 

33 

3 10857 

57 

4 

642877 

4 

3o 

953413    I 

o3 

689463 

5 

33 

3io537 

56 

5 

6431 35 

4 

3o 

953352    I 

o3 

689783 

5 

33 

310217 

55 

6 

643393 
643650 

4 

3o 

953290    I 

o3 

690103 

5 

33 

309897 

54 

7 

4 

29 

95322S    I 

o3 

690423 

5 

33 

309577 

53 

8 

643908 

4 

29 

953166    I 

o3 

690742 

5 

32 

309258 
30S938 

52 

9 

644165 

4 

29 

953104    I 

o3 

691062 

5 

32 

5i 

10 

644423 

4 

28 

953042    I 

o3 

691381 

5 

32 

308619 

5o 

II 

9-644680 

4 

28 

9-952980    I 

04 

9-691700 

5 

3i 

io.3o83oo 

8 

12 

644936 

4 

28 

952918   I 

04 

692019 

5 

3i 

307981 

i3 

645193 

4 

27 

952S55    I 

04 

69233s 

5 

3i 

30-662 

47 

14 

645430 

4 

27 

952793    I 

04 

692656 

5 

3i 

307344 

46 

i5 

645706 

4 

27 

952731    I 

04 

692975 

5 

3i 

307025 

45 

i6 

645962 

4 

26 

952669    I 

04 

693293 

5 

3o 

306707 

44 

\l 

646218 

4 

26 

952606    I 

04 

693612 

5 

3o 

3o6388 

43 

646474 

4 

26 

952544    I 

04 

693930 

5 

3o 

3o5o7o 

42 

19 

646729 

4 

25 

9524S1    I 

04 

694248 

5 

3o 

3o5752 

41 

20 

646984 

4 

25 

952419    I 

04 

694566 

5 

29 

3o5434 

40 

21 

9-647240 

4 

25 

9-9=i2356   I 

04 

9-694883 

5 

29 

io-3o5ii7 

ll 

22 

647494 

4 

24 

952294   1 

04 

695201 

5 

29 

3o4799 

23 

647749 

4 

24 

952231    I 

04 

695518 

5 

29 

3o44^2 

37 

24 

648004 

4 

24 

952168   I 

o5 

695836 

5 

11 

304164 

36 

25 

648258 

4 

24 

952106   I 

o5 

696153 

5 

3o3847 

35 

26 

648512 

4 

23 

952043   I 

o5 

696470 

.  5 

23 

3o3:')3a 

34 

27 

648766 

4 

23 

9519^^0   I 

o5 

696787 

5 

28 

3o32i3 

33 

28 

649020 

4 

23 

951917   I 

o5 

697103 

5 

28 

3o2«97 

32 

29 

649274 

4 

22 

95i854   1 

o5 

697420 

5 

27 

3o258o 

3i 

3o 

649527 

4 

22 

951791    I 

o5 

697736 

5 

27 

302264 

3o 

3i 

9-649781 

4 

22 

9-951728   I 

o5 

9-698053 

5 

27 

10.301947 

It 

32 

65oo34 

4 

22 

95i665   I 

o5 

698369 
698685 

5 

27 

3o 1 63 1 

33 

650287 

4 

21 

951602   I 

o5 

5 

26 

3oi3i5 

27  ! 

34 

65o539 

4 

21 

95 1 539   I 

o5 

699001 

5 

26 

'aM 

26 

33 

650792 

4 

21 

951476   I 

o5 

699316 

5 

26 

25 

36 

651044 

4 

20 

951412   I 

o5 

699632 

5 

26 

3oo368 

24 

37 

65 1 297 

4 

20 

951349   I 

06 

699947 

5 

26 

3ooo53 

23 

38 

65 1 549 

/  4 

20 

951286   I 

06 

700263 

5 

25 

299737 

22 

39 

65 1 800 

4 

19 

951222   I 

06 

700578 

5 

25 

299422 

21 

40 

652052 

4 

19 

951159   I 

06 

700S93 

5 

25 

299107 

20 

41 

9-652304 

4 

\l 

9-951096   I 

06 

9-701208 

5 

24 

10-298792 

\l 

42 

652555 

4 

95io32   I 

06 

701523 

5 

24 

29'^477 

43. 

652806 

4 

18 

95096S   I 

06 

701837 

5 

24 

298.63 

n 

44 

653o57 
6533o8 

4 

18 

950905   I 

06 

702152 

5 

24 

29^^43 

16 

45 

4 

18 

950841    I 

06 

702466 

5 

24 

297  ^'34 

i5 

46 

653558 

4 

17 

950778   I 

06 

702780 

5 

23 

29-!220 

14 

S 

6538o8 

4 

17 

950714   I 

06 

703095 

5 

23 

■2C)(x^o5 

i3 

654059 

4 

\l 

95o65o   I 

06 

703409 

5 

23 

296091 

12 

49 

654309 
654558 

4 

95o586   I 

06 

703723 

■  5 

23 

296277 

II 

5o 

4 

16 

95o522   I 

07 

704036 

5 

22 

295964 

10 

5i 

9.654808 

4 

16 

9-950458   I 

07 

9-704350 

5 

22 

io-29565o 

? 

52 

655058 

4 

16 

950394   I 

07 

704663 

5 

22 

295337 

^^ 

655307 

4 

i5 

95o33o   1 

07 

704977 

5 

22 

295023 

7 

i^ 

655556 

4 

i5 

950266   I 

07 

705290 

5 

22 

294710 

6 

55 

6558o5 

4 

i5 

950202   I 

07 

7o56o3 

5 

21 

294397 
294084 

5 

56 

656o54 

4 

14 

95oi38   I 

07 

705916 

5 

21 

4 

57 

656302 

4 

14 

950074   I 

07 

706228 

5 

21 

293712 

3 

58 

656551 

4 

14 

950010   I 

07 

706541 

5 

21 

293459 

2 

59 

656799 

4 

i3 

9499^^5   I 

07 

706854 

5 

21 

293146 

I 

60 

657047 

4-i3 

949881    I 

07 

707166 

5-20 

292834 

° 

Cosine 

D. 

Sine     T 

). 

Cotansr. 

D. 

Tansr. 

M. 

(63   DEGREES.) 


SIXES  AND  TANGENTS. 

(27  DEGREES.) 

45 

M.j 

Sii.e   1 

1). 

Codiiie  1 

1).   1 

Taw-,      i 

D.  1 

Cotung.  1 

""] 

0 

9-657047 

4-i3 

9.949881 

1-07  j 

9-707166 

5-20 

10-292834  I  60 

I 

637295 

4-. 3  1 

949*^1 'J 

1-07  j 

707478 

5-20  : 

292322  ;  59 

a 

657042 

4-12 

949732 

I  -07  1 

707790 

5.20  j 

292210  1  38 

3 

607790 

4-12 

949688 

I -08 

708102 

5-20 

291898  !  57 
291586  j  56 

4 

65»o37 

4-12   ! 

949623 

I- 08 

708414 

5-19 

5 

65-2S4 

4-12   1 

949558 

1-08 

708726 

5-19 

2912-4   55 

6 

63S53I 

4-11 

949494 

I -08 

709037 

5.19 

290963   54 

7 

6J8778 

4-11 

949429 

I -08 

709349 

^'9  1 

290651  1  53 

8 

639025 

4ii 

949364 

I -08 

709660 

i't 

290340  1  52 

9 

659271 

4-10 

949300 

I -08 

709971 

5-'5 1 

290029  ;  5i 

10 

659017 

4-10 

949235 

I -08 

710282 

5-i8 

289718  ;  5o 

II 

9-639763 

4-10 

9-949170 

I -08 

9.710593 

5.18 

10.289407  40 
289096  1  48 

12 

660009 

4-09 

949105 

1-08 

710904 

5-18 

i3 

660235 

4-09 

949040 

1-08 

711215 

5-i8 

288785 

47 

14 

660301 

4-09 

948975 

1.08 

711 525 

5-17 

288475 

46 

i5 

660746 

4-09 

948910 

I -08 

7II836 

5-17 

288164 

45 

i6 

660991 

4-o8 

948845 

1.08 

712146 

i-''' 

287834 

44 

17 

661236 

4-o8 

948780 

1-09 

712456 

5.17 

287544 

43 

18 

661481 

4-o8 

948715 

1.09 

712766 

5.16 

287234 

42 

•9 

661726 

4-07 

948650 

1.09 

718076 

5-16 

286924  1  41 

20 

661970 

4-07 

948584 

1-09 

713386 

5.16 

286614  40 

21 

9-662214 

4-07 

9-948519 

1-09 

9-713696 

5.16 

10-286304  39 
285993  38 

22 

662459 

4-07 

948454 

1.09 

714003 

5.16 

23 

662703 

4-o6 

948338 

1.09 

714314 

5-i5 

283686   37 
283376   36 

24 

662946 

4-o6 

948323 

1.09 

714624 

5-i5 

25 

663190 

4-o6 

948257 

1.09 

714933 

5-i5 

283067   35 

26 

663433 

4-o5 

948192 

1-09 

715242 

5-i5 

284758 

34 

27 

663677 

4-o5 

948126 

1-09 

7 1 555 1 

5.14 

2S4449 

33 

i  28 

663920 

4-o5 

948060 

I109 

715860 

5-14 

284140 

32 

29 

664163 

4-o5 

947995 

I-IO 

716168 

5-14 

283tJ32 

3i 

3o 

664406 

4-04 

947929 

I  -10 

716477 

5-14 

283323 

3o 

3i 

9.664648 

4-04 

9.947863 

I-IO 

9-716785 

5.14 

10. 28321 5 

^2 

32 

664891 

4-04 

947797 

I-lO 

717093 

5.13 

282907 

28 

33 

665 1 33 

4-o3 

9i-73i 

I. 10 

717401 

5-i3 

282J99 

27 

34 

665375 

4-o3 

947665 

1 .10 

717709 

5.i3 

282291 

26 

35 

665617 

4-o3 

947600 

I. 10 

7r8oi7 

5.i3 

2319S3 

23 

36 

665859 

4-02 

947533 

l-IO 

718323 

5-i3 

281670 

24 

ll 

666100 

4-02 

947467 

I. 10 

718633 

5.12 

281367 

23 

666342 

4-02 

947401 

I  .  10 

718940 

5-12 

281060 

23 

39 

666583 

4-02 

947335 

IIO 

719248 

5-12 

280732 

21 

40 

666824 

4-01 

947269 

I. 10 

719355 

5.12 

280445 

20 

4i 

9.667065 

4-01 

9-947203 

I. 10 

9-719S62 

5-12 

io.23oi38 

;? 

42 

667305 

4-01 

947136 

I-U 

720169 
720476 

5. II 

279831 

43 

667546 

4-01 

947070 

III 

5-II 

279324 

'1 

44 

667786 

4-00 

947004 

I  - 11 

720783 

5-II 

279217 

16 

45 

668027 

4-00 

946937 

1  - 1 1 

721089 

5. II 

27^911 

i5 

46 

668267 

4-00 

946871 

111 

721396 

5. II 

2-S604 

14 

47 

6685o6 

3-99 

946804 

I-U 

721702 

5.10 

278298 

i3 

48 

1   668746 

^■99 

946733 

I  -II 

722009 

5.10 

277991 

12 

49 

66S986 

^■99 

946671 

III 

722813 

5.10 

2776S5 

II 

5o 

669225 

3.99 

946604 

I-U 

722621 

5-10 

277379 

10 

5i 

9-669464 

3-98 

9-946538 

I-II 

9.722927 

5-10 

Io.2770^3 
276768 

I 

52 

669703 

3-98 

94647' 

III 

723232 

5-09 

53 

669942 

3-98 

946404 

I-II 

723538 

5-0.) 

276462 

I 

54 

670181 

3-97 

946337 

I-II 

723844 

5-09 

276136 

55 

670419 

3.97 

916270 

1-12 

724149 

3-09 

275831 

5 

56 

670658 

3.97 

946203 

112 

724454 

i   r°2 

275346 

4 

57 

670896 
671134 

3-97 

946136 

1-12 

724739 

1  5-o8 

273241 

3 

58 

3.96 

946069 

I  12 

725o63 

1  5.08 

274935 

2 

59 

671372 

3-96 

946002 

1-12 

725369 

5-08 

274631 

I 

66 

671609 

3.96 

945935 

1-12 

725674 

1  5.08 

274326 

0 

'  Cosine 

D. 

1   Sine 

T>. 

Cotanjr. 

!  D. 

Tangr. 

(62    DEGREES.) 


46 


:;3    DEGREES.)      A    TABLE    OF    LOGARITHMIC 


I-m: 

Sine 

D. 

Co-^uie 

D. 

Taiitr. 

D. 

Cotiuig. 

0 

9-671609 

3-96 

9-945935 

I- 12 

9-725674 

5-08 

10-274326 

60 

I 

671^47 

3 

95 

945^.63 

I  -12 

7259-9 

5 

08 

274021 

U 

2 

biio^Ji 

3 

95 

943800 

I  - 12 

7262^4 

5 

07 

273-716 

3 

672321 

3 

95 

943733 

1-12 

726388 

5 

07 

273112 

57 

4 

672558 

3 

95 

945666 

1-12 

726H92 

5 

07 

273 loS 

56 

5 

672795 
673o32 

3 

94 

943398 

I-I2 

727197 

5 

07 

272.03 

55 

6 

3 

94 

945531 

I  -12 

727501 

5 

07 

272499 

54 

7 
8 

673268 

3 

94 

945464 

I-l3 

727805 

5 

06 

272195 

53 

6735o5 

3 

9'» 

945396 

I-l3 

728109 

5 

06 

271S9, 

52 

9 

673741 

3 

93 

945328 

I-l3 

728412 

5 

06 

27I5S3 

5i 

lO 

673977 

3 

93 

945261 

i-i3 

728716 

5 

06 

271284 

5o 

II 

9-674213 

3 

93 

9-945193 

i-i3 

9-729020 

A 

06 

io-2-'0:;8o 

4Q 

12 

674448 

3 

92 

945i25 

i.i3 

729323 

5 

o5 

270677 

48 

i3 

674684 

3 

92 

945o58 

i-i3 

729626 

5 

o5 

270374 

47 

14 

674919 

3 

92 

944990 

i-i3 

729929 

5 

o5 

270071 

46 

i5 

675155 

3 

92 

944922 

i.i3 

730233 

5 

o5 

269761 

45 

i6 

675390 

3 

91 

944854 

1-13. 

73o535 

5 

o5 

269 ',65 

44 

\l 

675624 

3 

9' 

944786 

i-i3 

73o838 

5 

04 

269162 

43 

•  675859 

3 

91 

944718 

i-i3 

731141 

5 

04 

2688^9 

42 

»9 

676094 

3 

91 

944630 

i-i3 

731444 

5 

04 

268336 

41 

20 

676323 

3 

90 

944582 

1-14 

731746 

5 

04 

268234 

40 

21 

9-676562 

3 

90 

9-944514 

I-I4 

9-732048 

5 

04 

10-267932 

It 

22 

676796 

3 

90 

944446 

I-I4 

732351 

5 

o3 

267649 

23 

677030 

3 

^ 

944377 

I-I4 

732653 

5 

o3 

267347 

37 

24 

677264 

3 

944309 

1-14 

732933 

5 

o3 

267045 

36 

23 

677498 

3 

89 

944241 

I-I4 

733257 

5 

o3 

266743 

35 

26 

677731 

3 

89 

944172 

1-14 

733558 

5 

o3 

266442 

34 

27 

677964 

3 

88 

944104 

I-I4 

733860 

5 

02 

266140 

33 

28 

678197 

3 

88 

944036 

1. 14 

734162 

5 

02 

265838 

32 

29 

678430 

3 

88 

943967 

I -14 

734463 

5 

02 

265537 

3i 

3o 

678663 

3 

88 

943899 

I-I4 

734764 

5 

02 

265236 

3o 

3i 

9-678895 

3 

87 

9-943830 

I-I4 

9 -735066 

5 

02 

10-264934 

ll 

32 

679128 

3 

87 

943761 

I-I4 

735367 

5 

02 

264633 

33 

679360 

3 

87 

943693 

i-i5 

735668 

5 

01 

264332 

27 

34 

679592 

3 

87 

943624 

i-i5 

735969 

5 

01 

26403 1 

26 

35 

679^24 

3 

86 

943555 

i-i5 

736269 

5 

01 

263731 

25 

36 

68oo56 

3 

86 

943486 

i.i5 

736570 

5 

01 

263430 

24 

ll 

680288 

3 

86 

943417 

i-i5 

736871 

5 

01 

263 1 29 

23 

68o5i9 

3 

85 

943348 

ii5 

737171 

5 

00 

262829 

22 

39 

680750 

3 

85 

943279 

i-iS 

737471 

5 

00 

262529 

21 

40 

680982 

3 

85 

943210 

i-i5 

737771 

5 

00 

262229 

20 

41 

9.681213 

3 

85 

9-943141 

i-i5 

9-738071 

5 

00 

10-261929 

\l 

42 

681443 

3 

84 

943072 

i.i5 

733371 

5 

00 

261629 

43 

681674 

3 

84 

943003 

I  13 

738671 

4 

99 

261329 

\l 

44 

681905 

3 

84 

942934 

i-i5 

738971 

4 

99 

261029 

45 

682135 

3 

84 

942^64 

i-i5 

739271 

4 

99 

260729 

i5 

46 

682365 

3 

83 

942795 

i-i6 

739370 

4 

99 

260430 

14 

% 

682595 

3 

83 

942726 

i-i6 

739870 

4 

99 

26oi3o 

i3 

682825 

3 

83 

942636 

i-i6 

740169 

4 

99 

259831 

12 

49 

683o55 

3 

83 

942587 

i-i6 

740468 

4 

98 

259532 

II 

DO 

683284 

3 

82 

942317 

i-i6 

740767 

4 

98 

259233 

10 

5i 

Q.6835i4 

3 

82 

9-942448 

i-i6 

9.741066 

4 

98 

10-258934 

t 

52 

683743 

3 

82 

942378 

i-i6 

741365 

4 

98 

258635 

53 

683972 

3 

82 

9423o3 

1. 16 

741664 

4 

98 

258336 

I 

54 

684201 

3 

81 

942239 

1. 16 

741962 

4 

97 

258o38 

55 

684430 

3 

81 

942169 

i-i6 

742261 

4 

97 

257739 

5 

56 

684653 

3 

81 

942099 

1. 16 

742359 

4 

97 

257441 

4 

ll 

684887 

3 

80 

942029 

1-16 

742858 

4 

97 

237142 

3 

685ii5 

3 

80 

941939 

i-i6 

743 1 56 

4 

97 

256844 

2 

59 

685343 

3 

80 

9418.39 

'•17 

743454 

4 

97 

256546 

I 

60 

685571 

3.80 

941819 

I-I7 

743732 

4-96 

256248 

0 

Cosine 

D. 

Sine 

T). 

Cot:mnr. 

D. 

Tanir. 

(61    DEGREE^",.) 


SINES  AND  T.INGENTS.      (29    DEGREES.) 


47 


M. 

0 

Sine 

D.   1 

Cosine 

D.  1 

Tang.   I 

D.   1 

Cotang.     i 

9-685571 

3.80  ' 

9-941819 

1-17 

9-743752 

4.96  ; 

10-256248  60 

I 

685799 

3-79  ! 

941749 

I-I7 

744o5o 

4-96 

255930   5o 
255652   58 

a 

686027 

3-79  1 

941679 

I-I7 

74434^ 

4-96 

3 

686254 

3-79  1 

941609 

I-I7 

744645 

4.96 

255355  57  1 

4 

6B64S2 

n  \ 

941539 

I-I7 

744943 

4-96  1 

255o57 

56 

5 

686709 

941469 

1-17 

745240 

4.96 

254760 

55 

6 

686936  i 

3.78  1 

941398 

1-17 

745533 

4.93 

254462 

54  1 

I 

687163 

3.78  1 

941328 

1-17 

745835 

4-95 

254165 

53  ' 

■  687389  : 

3-78  ! 

941258 

I-I7 

746132 

4.95 

233868 

52  , 

9 

687616  ; 

3.77 

941 187 

I-I7 

746429 

4-9D 

233371 

5i 

10 

687843 

3-77 

941117 

1. 17 

746726 

4-95 

253274 

5o  i 

II 

9-688069 
688295 

3-77  I 

9-941046 

1.18 

9-747023 

4.94 

10-252977 

1?i 

12 

3-77  '• 

940975 

1-18 

747319 

4.94 

252681 

i3 

688521 

3-76  j 

940905 
940834 

i-,8 

747616 

4-94 

2523S4 

47  ' 

14 

688747 

3.76  I 

1-18 

747913 

4-94 

252087 

46  ; 

ID 

688972 

3-76  1 

940763 

1-18 

748209 
7485o5 

4-94 

251791 

45 

i6 

689198 

3.76  1 

940693 

i-i8 

4-93 

25149^ 

44 

n 

689423 

3.75  i 

940622 

1-18 

748801 

4-93 

25II99 
25o9o3 

43 

i8 

689648 

3.75  1 

94o55i 

i-i8 

749097 

4.93 

42 

19 

689873 

3.75  1 

940480 

I -18 

7493o3 
749689 

493 

25o6o7 

41 

20 

690098 

3.75  ' 

940409 

i-i8 

4.93 

25o3ii 

40 

21 

9-690323 

3-74 

9-940338 

1.18 

9.749983 

4-93 

io-25ooi5 

^ 

22 

690548 

3-74 

940267 

i-i8 

750281 

4-92 

249719 

23 

690772 

3.74  ' 

940196 

1-18 

700376 

4-92 

249424 

37 
36 

24 

690996 

3-74  1 

940125 

1-19 

730872 

4-92 

249128 

23 

691220 

3-73  : 

940054 

1-19 

731167 

4-92 

248833 

35 

26 

691444 

3.73  i 

939982 

1-19 

731462 

4-92 

248538 

34 

11 

691668 

3.73  ! 

93991 1 

I-I9 

751757 

4.92 

248243 

33 

691892 

3.73 

939840 

1-19 

752o52 

4-91 

247948 

32 

29 

692115 

3.72 

939168 

1-19 

752347 

4-91 

247653 

3i 

3o 

692339 

3-72 

939697 

I- 19 

752642 

4-91 

247358 

3o 

3i 

9-692562 

3.72  ! 

9-939625 

I -19 

9-752937 

4.91 

10- 247063 

It 

32 

692785 

3-71   ; 

939554 

1-19 

73323i 

4-91 

246769 

33 

693008 

3.71   ] 

939482 

1-19 

753526 

4.91 

246474 

11 

34 

693231 

3.71 

939410 

1-19 

753820 

4.90 

246180 

35 

693453 

3.71 

939339 

I-I9 

754115 

4-90 

245885 

25 

36 

693676 

3.70  ! 

939267 

1-20 

754409 
754703 

4.90 

245591 

24 

37 

693898 

3.70  i 

939195 

I  -20 

4.-90 

245297 

23 

38 

694120 

3-70  1 

939123 

1-20 

754997 

4.90 

245oo3 

22 

39 

694342 

3-70 

939032 

1-20 

755291 

4-90 

244709 
244415 

21 

40 

694564 

3-69  ! 

93S9S0 

1-20 

755585 

4.89 

20 

41 

9-694786 

3.69 

9-938908 

1-20 

9-755878 

4-89 

10-244122 

\l 

42 

695007 

3-69 

938836 

1-20 

756172 

4.89 

243S28 

43 

695229 

3-69 

93S763 
93S691 

1-20 

736465 

4.89 

243535 

7 

44 

695450 

3-68 

1-20 

736759 

4-89 

243241 

16 

45 

693671 

3-68 

938619 

1-20 

737032 

4-?? 

242948 

i5  j 

46 

695892 

3-68 

938547 

1-20 

757345 

4.88 

242655 

'i  ' 

47 

!   6961 i3 

3-68 

938475 

1-20 

737638 

HI 

242362 

i3  ' 

4-S 

690334 

3-67 

938402 

I-2I 

737931 

4-?^ 

242069 

12 

49 

696554 

3-67 

938330 

I-2I 

758224 

4-^? 

241776 

11 

DO 

696775 

3-67 

938258 

I-2I 

758517 

4.88 

241483 

10 

5i 

9 • 696995 

III 

9-938185 

I-2I 

9-758810 

4.88 

10-241190 

t 

52 

697215 

9381 13 

I-2I 

739102 

4-87 

240898 

53 

697435 

3-66 

938040 

I-2I 

759305 
7396S7 

4-87 

240603 

I 

54 

697654 

3-66 

937967 

1-21 

4-87 

•   24o3i3 

55 

697874 

3-66 

937895 

I-2I 

759979 

4-87 

24002 1 

5 

56 

698094 

3-65 

937822 

I-2I 

760272 

4.87 

239728 

4 

57 

698313 

3-65 

937749 

I-2I 

760364 

Hi 

239436 

3 

58 

698532 

3-65 

937676 

I-2I 

760856 

4-86 

239144 

2 

•59 

698751 

3^65 

937604 

I-2I 

761.48 

4-86 

238852 

I 

6o 

698970 

3.64 

937531 

1-21 

761439 

4-86 

j   a3856i 

0 

Cosine 

D. 

'   Sine 

D. 

Cotans?. 

D. 

1  Tang. 

"nT 

S7 

(60 

DEQR 

EES.) 

48 


^30    DEGREES.;     A   TABLE   OF   LOGARITHMIC 


0 

Siiie 

D. 

Cosine  ..      D. 

Tang. 

D. 

Cotang. 

9-698970 

3.64 

9-937531   1 

-21 

9-761439 

4-86 

10- 238561 

60 

I 

699189 

3.64 

937458   I 

-22 

76,731 

4-86 

238269 

2 

699407 

3.64 

937385  ,  I 

-22 

762023 

4-86 

237977 

3 

699626 

3-64 

937312  1  1 

.22 

762314 

4-86 

237686 

57 

4 

699844 

3.63 

937238  !   I 

•22 

762606 

4-85 

237394 

56 

5 

700062 

3.63 

937165  !  I 

•22 

&l 

4-85 

237103 

55 

6 

700280 

3-63 

937092  1  I 

•22 

4-85 

2368,2 

54 

7 

700498 

3-63 

937019  j  1 

-22 

763479 

4-85 

23652, 

53 

8 

700716 

3-63 

936946  I  I 

•22 

763770 

4-85 

236230 

52  ! 

9 

700933  1 

3-62 

936872  1  I 

•22 

764061 

4-85 

235939 
235648 

5, 

10 

70ii5i 

3-62 

936799   I 

.22 

764352 

4-84 

5o 

1 
1  II 

9.701368 

3-62 

9.936725  :  1 

22 

9-764643 

4.84 

10.235357 

\^ 

12 

70i585 

3-62 

936652   I 

•23 

764933  ~ 

4-84 

235067 

i3 

701802  i 

3.61 

936078   I 

23 

763224 

4-84 

234776 

47 

U 

702019 
702236 

3.61 

9365o5   I 

23 

7655,4 

4-84 

234486 

46 

i5 

3.61 

936431    I 

23 

7658o5 

4-84 

234195  1  45  1 

i6 

702452  j 

3.61 

936357    I 

23 

766095 

4-84 

233905 
23361 5 

1  44 

^2 

702669 
702885  1 

3.60 

936284  ,  I 

23 

766385 

4-83 

43 

i8 

3 -60 

936210  1  1 

23 

766675 

4-83 

233325 

!  42 

19 

7o3ioi  ' 

3.60 

936i36  ■  1 

23 

766965 

4-83 

233o35 

1  41 

20 

703317 

3.60 

936062   I 

23 

767255 

4-83 

232745 

1  40 

21 

9-703533  1 

3.59 

9.935988  1  1 

23 

9.767545 

4-83 

10.232455 

U 

22 

703749  1 

3-59 

9350,4   I 
933840   I 

23 

767834 

4-83 

232,66 

23 

703964 

3.59 

23 

768124 

4-82 

23,876 

37 

24 

704179 
704395  i 

3.59 

935766   1 

24 

768413 

4-82 

23,587 

36 

25 

lU 

935692  '  1 

24 

768703 

4-82 

23,297 

.35 

26 

704610 

935618  :  1 

24 

768992 

4-82 

23 1 008 

34 

^2 

704825 

3-58 

935543  1 

24 

769281 

4-82 

2307,9 

33 

7o5o4o 

3-58 

935469  I 

24 

769570 

4-82 

23o43o 

32 

29 

705254 

3.58 

935395  I 

24 

769860 

4-8i 

23o,4o 

3, 

3o 

705469  , 

•3.57 

935320  I 

24 

770148 

4-8i 

22qS52 

3o 

3i 

9-705633  I 

3.57 

9.935246  i  I 

24 

9.770437 

4-8, 

10.229563 

.1 

32 

705898 

3.57 

935171  1  1 

24 

770726 

4 -81 

220274 
228985 

33 

7061 1 2 

3-57 

935097  :   I 

24 

77,0,5 

4-8i 

27 

34 

706326 

3-56 

935022  ;  I 

24 

77i3o3 

4-8i 

22S697 

26 

35 

706539 

3-56 

934948  I  I 

24 

771592 

4-8i 

22S408 

25  1 

36 

706753 

3.56 

934873  I  I 

24 

771880 

4-8o 

228,20 

24  i 

?7 

706967 

3-56 

93479^  I  I 

25 

772,68 

4-8o 

227832 

23 

38 

707180 

3-55 

934723  ;   I 

25 

772457 

4-8o 

227543 

22 

39 

707393 

3.55 

934649  '  I 

25 

772745 

4-8o 

227255 

21 

40 

707606  ' 

3.55 

934574     I 

25 

773o33 

4- 80 

226967 

20  1 

41 

9.707819  ' 

3.55 

9-934499  » 

25 

9.773321 

4-8o 

10.226679 

;?i 

42 

708032 

3.54 

934424  !  I 

25 

773608 

4-79 

226392 

43 

708245 

3.54 

934349  1  ' 

25 

773806 
774184 

4-79 

226104 

17  i 

44 

708458  : 

3.54 

934274  !  1 

25 

4-79 

225Si6 

,6 

45 

708670 
708882 

3.54 

934199   I 
934123  1  I 

25 

774471 

4-79 

225529 

i5 

46 

3.53 

25 

774739 

4-79 

225241 

14 

^l 

709094 

3.53 

934048  1  1 

25 

775046 

4-79 

224954 

i3 

48 

709306 

3.53 

933973  !  I 
933898  !  1 

25 

775333 

4-79 

224667 

12 

49 

709518 

3.53 

26 

77562, 

4-78 

224379 

II 

5o 

709730 

3.53 

933822   I 

26 

775908 

4-78 

224092 

10 

5i 

9.709941  I 

3-52 

9933747   I 

26 

9-776,95 
776482 

4-78 

io.2238o5 

? 

52 

710153  1 

3.52 

933671   I 

26 

a-78 

2235,8 

53 
54 

7io364 
710575 

3-52 
3.52 

933596   1 
933520  1  I 

26 
26 

]lpoti 

4-78 
4-78 

22323, 

222945 

7 
6 

55 

710786 

3.5i 

933445   I 

26 

777342 

4-78 

222658 

5 

56 

710997 

3.5i 

933369  ,  1 

26 

777628 

4-77 

222372 

4 

ll 

711208 

3.51 

933293  I  1 

26 

7779,5 

4-77 

222085 

3 

711419 

3.5i 

933217   I 

26 

778201 

4-77 

22,799 

2 

^ 

711629 

3-50 

933141   I 

26 

778487 

4-77 

22,5,2 

I 

60 

711839  ; 

3.50 

933066   1 

26 

778774 

4-77 

221226 

M.| 

Cosino  1 

D. 

Sine   1  I 

). 

Cotans. 

D. 

Tang. 

(50  DEGREES.) 


SIXES   AXD   TAXGENTS.      (31    DEGREES.) 


49 


M^l 

Sine 

D. 

Cosine 

D. 

Tang,   i 

D.   1 

Cotansr.  i 

0 

9.711839 

3-5o  1 

9'933o66  j 

1-26 

9-778774 

4-77 

10-221226 

60 

I 

7120D0 

3.50 

932990 

1-27 

779060  1 

4-77 

220940 

59 

2 

712260 

3-50 

932914 

1.27 

779346 

4.76 

220654 

58 

3 

712469 

3-49 

932838 

1-27 

779632 

4-76 

220868 

57 

4 

-712679 

3-49 

932762 

1.27 

779918 

4-76 

220082 

56 

5 

712889 

3-49 

9826^5 

1-27 

780203 

4-76 

219797 

55 

6 

7.3098 

3-49  1 

932609 

1-27 

780489 

4-76 

219511 

54 

7 

7i33o8 

3-49 

932533 

1-27 

780775 

4-76 

219225 
218940 

53 

8 

7i35i7 

3-48 

932457 

1-27 

781060 

4-76 

52 

9 

713726 

3.48 

932380 

1-27 

781846 

4.75 

218654 

5i 

10 

713935 

3.48 

932304 

1-27 

781681 

4-75 

218869 

5o 

II 

9-714144 

3-48 

9-932228 

1.27 

9-781916 

4-75 

10-218084 

% 

12 

714352 

3-47 

93oi5i 

1-27 

782201 

4-75 

217799 

i3 

714561 

3-47 

982075 

1-28 

782486 

4-75 

217514 

47 

14 

714769 

3-47 

931998 

1-28 

782771 

4-75 

217229 

46 

i5 

714978 

3-47 

981921 

1-28 

783o56 

4-75 

216944 

45 

i6 

7i5i86 

3-47 

93.845 

1-28 

788841 

4-75 

216659 

44 

•7 
i8 

715394 

3-46 

981768 

1-28 

788626 

4-74 

216874 

43 

7i56o2 

3-46 

981691 

1-28 

788910 

4-74 

216090 

42 

'9 

715809 

3-46 

981614 

1-28 

784195 

4-74 

2i58o5 

41 

20 

716017 

3-46 

981537 

1.28 

784479 

4-74 

215521 

40 

21 

9-716224 

3-45 

9-931460 

1.28 

9-784764 

4-74 

10-215236 

ll 

22 

716432 

3-45 

981883 

1.28 

785048 

4-74 

214952 

23 

7 1 6639 

3-45 

981806 

1.28 

785882 

4-73 

214668 

37 

24 

716846 

3-45 

981229 

1-29 

7856i6 

4-73 

214384 

36 

25 

717053 

3.45 

93ii52 

1-29 

785900 

4-73 

214100 

35 

26 

-C7259 

3-44 

981075 

1-29 

786184 

4-73 

2i38i6 

34 

1  27 

717466 

3-44 

980998 

1-29 

786468 

4-73 

213582 

33 

28 

717673 

3-44 

980921 
980843 

1.29 

786752 

4-73 

218248 

32 

^9 

7i7«79 

3-44 

1-29 

787086 

4-73 

212964 

3i 

3o 

718085 

3.43 

980766 

1.29 

787319 

4-72 

212681 

3o 

3i 

9-718291 

3-43 

9-980688 

1-29 

9-78^608 

4-72 

10-212897 

^ 

32 

718497 

3.43 

980611 

1.29 

787886 

4-72 

212114 

33 

718703 

3-43 

93o533 

1-29 

788no 

4-72 

2ii83o 

27 

34 

718909 

3.43 

9I0456 

1.29 

788453 

4-72 

211547 

26 

35 

719114 

3-42 

980878 

1-29 

788786 

4-72 

211264 

25 

36 

719320 

3.42 

930800 

1.80 

789019 

4-72 

2 1 098 1 

24 

37 

719525 

3-42 

930228 

1-30 

789802 

4-71 

210698 

23 

38 

719730 

3-42 

980145 

1.30 

789585 

4-71 

2io4i5 

22 

39 

719935 

3-41 

980067 

1.80 

7H9868 

4-71 

210182 

21 

4o 

720140 

3-41 

929989 

i-3o 

790i5i 

4-71 

209849 

20 

4« 

9-720345 

3-41 

9-920911 

1.30 

9-790438 

4-71 

10-209567 

\l 

42 

720549 

3-41 

929S33 

I -30 

790716 

4-71 

209284 

43 

720754 

3-40 

929755 

i.3o 

790999 

4-71 

209001 
208719 

17 

44 

72095s 

3-40 

929677 

1-30 

79.281 

4-71 

16 

45 

721162 

3-40 

919'^m 

1-80 

79.563 

4-70 

208487 

i5 

46 

721 366 

3.40 

929521 

i.3o 

791846 

4-70 

2o8i54 

14 

47 

72070 

3-40 

929442 

1-30 

792128 

4-70 

207872 

i3 

48 

721774 

3-39 

929864 

I-8I 

792410 

4-70 

207590 

12 

49 

721978 

3.39 

929286 

i-3i 

792692 

4-70 

207808 

II 

5o 

722181 

3-39 

929207 

I-3I 

792974 

4-70 

207026 

10 

5i 

9-722385 

3.39 

9-929129 

1.31 

9-798256 

4-70 

10-206744 

% 

52 

722583 

3.39 

929050 

1-31 

798538 

4-69 

206462 

53 

-722791 

3-38 

92S972 

1.31 

7938.9 

4-69 

206181 

I 

54 

722994 

3-38 

92SM93 

1-31 

79iioi 

469 

205899 

55 

723197 

3-38 

92S815 

i.3i 

794383 

4-69 

2o56i7 

56 

723400 

3-38 

928736 

i-3i 

794664 

4-69 

205336 

57 

7236o3 

3-37 

928657 

i-3i 

794945 

4-69 

2o5o55 

58 

7238o5 

3.37 

928578 

i.3i 

795227 

4-69 
4-68 

204773 

59 

724007 

3-37 

928499 

i-3i 

795508 

204492 

6o 

724210 

3-37 

928420 

i-3i 

795789 

4-68 

2042 II 

Cosine 

D. 

Sine 

I). 

1  Cotansr. 

D. 

TanjT. 

M. 

(58    DEGREES.) 


50 


(32   DEGREES.)      A  TABLE   OF   LOGARITHMIC 


^L 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang.  i 

0 

7.724210 

3.37 

9.928420 

,-32 

9.795789 

4-68 

10-2042.1  1  60  } 

I 

724412 

3.37 

928342 

1-32 

796070 

4-68 

203930   59  i 
203649   58 

2 

724614 

3.36 

928263 

1-32 

796351 

4-b8 

3 

724816 

3.36 

928183 

1-32 

796632 

4-68 

203368  ;  57 

4 

725017 

3-36 

928104 

1.32 

796913 

4-68 

203087  i  56 

5 

720219 

3-36 

92S025 

1.32 

797194 

4.68 

202806   55 

6 

725420 

3.35 

927946 

1.32 

797473 

4-68 

2O2025  :  5i 

7 

725622 

3.35 

927867 

1.32 

797733 

4-68 

202245  :  53 

8 

725823 

3-35 

927787 

1.32 

798036 

4-67 

201964   52 

9 

726024 

3-35 

927708 

1.32 

7983.6 

4-67 

20.684  i  5i 

■'o 

726225 

3.35 

927629 

1.32 

798596 

4-67 

20.404  ;  5o 

II 

9.726426 

3.34 

9.927549 

1.32 

9-798877 

4-67 

IO-201I23  j  40 

200843   48 

12 

726626 

3.34 

927470 

1-33 

799157 

4-67 

i3 

726S27 

3-34 

927390 

1.33 

799437 

4-67 

200563 

47 

14 

727027 

3.34 

927310 

1-33 

799717 

4-67 

200283 

46 

ID 

727228 

3-34 

927231 

1-33 

799997 

4-66 

2oooo3 

45 

i6 

727428 

3-33 

927101 

1-33 

800277 

4-66 

199723 

44 

17 

727628 

3.33 

927071 

1-33 

800557 

4-66 

199443 

43 

18 

727828 

3-33 

926991 

1-33 

8oo836 

4-66 

',^ti 

42 

19 

728027 

3-33 

92691 1 
926831 

1-33 

801116 

4-66 

41 

20 

728227 

3.33 

1-33 

801396 

4-66 

198604 

40 

21 

9-728427 

3-32 

9-926751 

1-33 

9-801675 

4-66 

10-198325 

39 

22 

728626 

3.32 

926671 

1-33 

801905 

4-66 

198045 

38 

23 

'  728825 

3.32 

926091 

1-33 

802234 

4-65 

197766 

37 

24 

729024 

3.32 

9265ii 

1-34 

8n35l3 

4-65 

197487 

36 

25 

729223 

3-31 

926431 

1-34 

&02'792 

4-65 

197208 

35 

26 

729422 

3.31 

926301 

1-34 

803072 

4-65 

196928 

34 

27 

729621 

3.31 

926270 

1-34 

8o335i 

4-65 

196649 

33 

23 

729820 

3-3i 

926190 

1-34 

8o363o 

4-65 

196370   32 

29 

730018 

3.30 

926110 

1-34 

803903 

4-65 

196092   3 1 

3o 

730216 

3.30 

926029 

1-34 

804.87 

4-65 

195813   3o 

3i 

9-730415 

3.30 

%IUU 

-34 

9-804466 

4-64 

10.195534 

It 

32 

73o6i3 

3.30 

8.-.V740 

4-64 

195255 

33 

73081 1 

3.30 

9207S8 

1-34 

800023 

4-64 

194977 

27 

34 

731009 

3.29 

925707 

1-34 

?oo3o2 

4-64 

194698 

26 

35 

731206 

3.29 

920626 

1.34 

800080 

4-64 

194420 

25 

36 

731404 

3.29 

925545 

1-35 

800809 

4-64 

194.41 

24 

37 

731602 

3.29 

920465 

1-35 

806.37 

4-64 

193863 

23 

38 

731799 

3.29 

925334 

1.35 

8064 .5 

4-63 

193585 

22 

39 

731996 

3.28 

9253o3 

1-35 

806693 

4-63 

193307 

21 

40 

732193 

3.28 

920222 

1.35 

80697 1 

4-63 

193029 

20 

41 

9.732390 
7325§7 

3.28 

9-920141 

1.35 

9-807249 

4-63 

10.192751 

\t 

42 

3-28 

920060 

1-35 

807527 

4-63 

192473 

43 

732784 

3.28 

924979 

1-35 

807805 

4-63 

192195 

17 

44 

732980 

3.27 

924897 

1-35 

8080.33 

4-63 

191917 

.6 

45 

733177 

3-27 

924816 

1-35 

8o836i 

4.63 

191639 

.5 

46 

733373 

3-27 

924735 

1-36 

8o8638 

4.62 

19.362 

14 

47 

733569 

3-27 

924654 

1-36 

808916 

4-62 

191084 

i3 

48 

73376D 

3-27 

924572 

1-36 

809.93 

4.62 

190807 

12 

49 

733961 

3-26 

924491 

1-36 

809471 

4.62 

190529 

II 

5o 

734157 

3-26 

924409 

1-36 

809748 

4-62 

190252 

10 

5i 

9.734353 

3.26 

9-924328 

1.36 

9-8.0025 

4-62 

10.189975 

t 

52 

734549 

3-26 

924246 

1-36 

8.o3o2 

4-62 

189698 

53 

734744 

3.25 

924164 

1-36 

8io58o 

4-62 

189420 

I 

54 

734939 

3.25 

9240.33 

1-36 

810857 

4-62 

189.43 

55 

735i35 

3.25 

924001 

1-36 

8iii34 

4-61 

188866 

5 

56 

735330 

'  3. 25 

923919 

1-36 

811410 

4-6i 

188590 

4 

57 

735525 

'   3-23 

923837 

1.36 

811687 

4-61 

i883i3 

3 

58 

735719 

3.24 

923755 

1.37 

811964 

4-6i 

i88o36 

2 

59 

7359.4 

3.24 

9236-3 

1.37 

812241 

4-61 

187759 
187483 

I 

60 

736109 

3.24 

923091 

1.37 

8i25i7 

4-61 

0 

Cosine 

r». 

Sine 

TV. 

('..tansr. 

D. 

Tanff. 

M. 

(57  DEGREES.) 


SINES  AND   TANGENTS       (33    DEGREES.) 


51 


M.| 
o 

Sine 

D. 

Cosine  | 

D. 

Tan-. 

D. 

Cotang.  j 

9.736109 

3-24  ^ 

9-923591 

1.37 

9.8125.7 

4-61  i 

10-187482  ! 

60 

I 

7363o3 

3-24 

923309 

1-37  : 

812794 

4-61  j 

187206  1 

i? 

2 

736498 

3-24  1 

923427 

1.37  1 

8i3o7o 

4-61 

186930  ! 

3 

736692 
736886 

3-23  1 

923345 

1.37  ! 

813347 

4-60 

186653  ; 

57 

4 

3-23 

923263 

1.37  1 

8i3623 

4-60 

186377  i 

56 

5 

737080 

3-23 

923181 

1.37 

813899 

4-60 

186101  . 

55 

6 

737274 

3-23 

928098 

1.37 

814175 

4.60 

I8582D 

54  i 

7 

737467 

3-23  i 

923016  1 

1.37 

814452 

4-60 

185548 

53 

8 

737661 

3-22   ; 

922933 

1-37 

814728 

4-60 

185272 

52 

9 

737855 

3-22 

922851 

1-37 

8i5oo4 

4-6o 

184996 

5i 

10 

738048 

3-22 

922768 

1-38  i 

815279 

4-6o 

184721 

DO 

1 1 

9-738241 

3-22 

9-922686 

1-38  1 

9-815555 

4-59 

10-184445 

it 

12 

738434 

3-22 

922603 

1-38 

8i583i 

4-59 

184169 

i3 

738627 

8-21 

922320 

1-38  i 

816107 

4-59 

183893 

47 

14 

738820 

3-21 

922438 

1-38 

8i6382 

4.59 

i836i8 

46 

ID 

739013 

3-21 

922355 

1-38 

8 16658 

4-59 

1833^2 

45 

i6 

730206 

3-21 

922272 

1-38 

816933 

4-39 

183067 

44 

17 

739398 

3-21 

922189 

1-38 

817209 

4-59 

182791 

43 

18 

739390 

3 -20 

922106 

1-38 

817484 

4-59 

i825i6 

42 

19 

739783 

3-20 

922023 

1-38 

817759 

4-50 
4-58 

182241 

41 

20 

739973 

3-20 

921940 

1-38 

8i8o3d 

181965 

40 

21 

9-740167 

3.20 

9-921857 

.-39 

9-8iS3io 

4-58 

10-181690 

li 

22 

740359 

3-20 

921774 

1.39 

8 1 8585 

4-58 

181415 

23 

74o35o 

3-19 

921691 

1-39 

818860 

4-58 

181140 

11 

24 

740742 

3.19 

921607 

1-39 

819135 

^1^ 

i8o865 

25 

740934 

3-19 

921524 

1.39 

819410 

4-58 

180590 

35 

26 

741125 

3.19 

921441 

1-39 

819684 

4-58 

i8o3i6 

34 

27 

74i3i6 

3-10 
3.18 

921337 

1-39 

819959 

4.58 

180041 

33 

28 

74i5o8 

921274 

1-39 

820234 

4.58 

179766 

32 

29 

741699 

3-18 

921190 

1-39 

82o5o8 

4-57 

179492 

3i 

3o 

741889 

3.18 

921107 

1-39 

820783 

4.57 

179217 

3o 

3i 

9-742080 

3.18 

9-921023 

1.39 

9-821057 

4.57 

10.178943 

It 

32 

742271 

3-18 

920939 

1-40 

821332 

4-57 

178668 

33 

742462 

3-17 

920856 

1-40 

821606 

4.57 

178394 

11 

34 

742652 

3.17 

920712 

1-40 

821880 

4.57 

178120 

33 

742842 

3.17 

920688 

1-40 

822154 

4.57 

177846 

25 

36 

743o33 

3-17 

920604 

1-40 

822429 

4.57 

177571 

24 

37 

743223 

3.17 

920520 

1-40 

822703 

4.57 

177297 

23 

38 

743413 

3.16 

920436 

1-40 

822977 

4-56 

177023 

22 

39 

743602 

3.16 

920332 

1-40 

8232  30 

4-56 

176750 

21 

40 

743792 

3-i6 

920268 

1-40 

823524 

4-56 

176476 

20 

41 

9-743982 

3-16 

9-920184 

1-40 

9-823798 

4-56 

10-176202 

\l 

42 

744171 

3.16 

920092 

1-40 

814072 

4-56 

173928 

43 

744361 

3. ID 

920013 

1-40 

824345 

4-56 

175633 

'1 

44 

744530 

3-i5 

919931 

1-41 

824c^io 

4-56 

175381 

16 

45 

744739 

3-i5 

919S46 

1-41 

8248^3 

4-56 

175107 

i5 

46 

744928 

3-i5 

919762 

1-41 

823166 

4-56 

174834 

14 

47 

743117 

3i5 

919677 

1-41 

825439 

4-55 

174361 

i3 

48 

745306 

3-14 

919393 

1-41 

823713 

■  4-55 

1742S7 

12 

I  49 

743404 

3.14 

[   919308 

I-4I 

823986 

1  4-55 

174014 

II 

5o 

745683 

3-14 

919I24 

1-41 

826239 

,  4-^)5 

173741 

10 

5i 

9-745871 

3-14 

9-919339 

1-41 

9-826532 

,-55 

tio- 173468 

t 

52 

746059 

3-14 

919234 

1-41 

826805 

4-5I> 

1   173195 

53 

746248 

3i3 

919169 

1-41 

827078 

4-55 

1    I72Q22 

7 

54 

746436 

4-i3 

9i90'<5 

1-41 

827331 

4-5!. 

172649 

6 

55 

746624 

3-i3 

919000 
918015 
918830 
918745 

1-41 

827624 

4-55 

!     172376 

5 

56 

746812 

3-i3 

1-42 

827897 

4-54 

•    172103 

4 

57 
58 

746999 
747187 

3-i3 

3-12 

1-42 
1-42 

828170 
828442 

4-54 
4-54 

1     171830 

1   ''^'^il 

3 
2 

59 

747374 

3-12 

918639 

1-42 

828715 

4-54 

17J285 

1  I 

66 

747362 

3-12 

918574 

1-42 

SiSg^l 

4-54 
.   D. 

1   171013  '  0 
1  Tang.   V^. 

Cosine 

1   D. 

i   Sine 

D. 

1  Cotancr. 

i 

8 

(56 

DEGR 

EES.) 

52 


(34   DEGREES.)     A  TABLE  OF  LOGARITHMIC 


f^iT 

Sine 

D. 

Cosine 

1  ^' 

Tiing. 

D. 

Cotang. 

0 

9-747562 

3.12 

9-918574 

1.42 

9-828987 

4.54 

10-171013 

60  i 

I 

747749 

3 

•12 

918489 

1-42 

829260 

4-54 

170740 

u\ 

2 

747936 
748123 

3 

•  12 

918404 

1-42 

829532 

4-54 

170468 

3 

3 

II 

9i83i8 

1.42 

829805 

4-54 

170195 

57  i 

4 

748310 

3 

II 

918233 

1.42 

830077 

4-54 

169923 

56 

5 

748497 

3 

II 

Q18147 

1.42 

83o349 

4-53 

169651 

55 

6 

748683 

3 

II 

918062 

1-42 

83o62. 

4-53 

169879 

54  I 

7 

748870 

3 

11 

917976 

1-43 

830893 

4.53 

169.07 

58 

8 

749056 

3 

10 

9 1 789 1 

1.43 

831.63 

4-53 

168885 

52 

9 

749243 

3 

10 

917805 

1.43 

831437 

4-53 

168563 

5i 

10 

749429 

3 

10 

917719 

1-43 

831709 

4-53 

168291 

!yo 

II 

9-749615 

3 

10 

9-917634 

1.43 

9-83i98i 

4-53 

10-168019 

49 

12 

749801 

3 

10 

9.7548 

1.43 

832253 

4-53 

167747 
167475 

48  j 

i3 

749987 

3 

09 

917462 

1-43 

832525 

4-53 

47 

U 

750172 

3 

09 

917376 

1.43 

l^.Vt 

4-53 

167204 

46 

i5 

75o358 

3 

09 

917290 

1-43 

833o68 

4.52 

166982 

45 

i6 

75o543 

3 

09 

917204 

1-43 

833339 

4-52 

1 6666 1 

44 

17 

750729 

3 

09 

917118 

1-44 

8336.1 

4-52 

166889 
1661.8 

43 

i8 

730914 

3 

08 

917082 

1-44 

8338.82 

4-52 

42 

19 

751099 
751284 

3 

08 

916946 
916859 

1-44 

834.54 

4-52 

165846 

41  1 

20 

3 

08 

1-44 

834425 

4-52 

165575 

40 

21 

9-751469 

3 

08 

9.916773 

1-44 

9-834696 

4-52 

io-i653o4 

39 

22 

^  751654 

3 

08 

9.6687 

1-44 

834967 

4-52 

i65o88 

38 

23 

751839 

3 

08 

916600 

1-44 

835238 

4-52 

164762 

37 

24 

752023 

3 

07 

9i65i4 

1-44 

835509 

4-52 

164491 

36 

25 

752208 

3 

07 

916427 

1-44 

8.35780 

4-51 

164220 

35 

26 

752392 

3 

07 

9.6341 

1-44 

836o5i 

4-5i 

168949 
168678 

34 

27 
28 

752576 

3 

07 

916254 

1-44 

836322 

4-51 

33 

752760 

3 

07 

916167 

1.45 

836593 

4-51 

168407 

32 

29 

752944 

3 

06 

9.6081 

1.45 

836864 

4-5i 

163.36 

3i 

3o 

753128 

3 

06 

915994 

1-45 

837.34 

4-51 

162866 

3o 

3i 

9.753312 

3 

06 

9-915907 
9.5820 

1.45 

9-837405 

4.51 

10-162595 

29 

32 

753495 

3 

06 

1.45 

837675 

4-5i 

162825 

28 

33 

753679 

3 

06 

9.5733 

1-45 

837946 

4-5i 

162054 

27 

34 

753862 

3 

b5 

9.5646 

1.45 

8382.6 

4-5i 

16.784 

26 

35 

754046 

3 

o5 

9.5559 

1.45 

888487 

4.50 

i6i5.3 

25 

36 

754229 

3 

o5 

9.5472 

1-45 

888757 

4-5o 

16.243 

24 

37 

754412 

3 

o5 

9.5385 

1-45 

889027 

4-5o 

160978 

23 

38 

754595 

3 

o5 

9.5297 

1-45 

o?9?97 

4-5o 

160708 

22 

39 

754778 

3 

04 

9.5210 

1-45 

889568 

4-5o 

160482 

21 

40 

754960 

3 

04 

9.5123 

1-46 

839888 

4-50 

160162 

20 

41 

9-755143 

3 

04 

9-9.5035 

1.46 

9-840108 

4-5o 

10-159892 

;i 

42 

755326 

3 

04 

914948 

1-46 

840878 

4-5o 

159622 

43 

755508 

3 

o4 

914860 

1.46 

840647 

4-5o 

159353 

17 

44 

755690 

3 

04 

914773 

1-46 

840917 

4.49 

159088 

i588i3 

16 

45 

755872 

3 

o3 

9.46S5 

1-46 

84.187 

4-49 

i5 

46 

756054 

3 

o3 

914598 

1-46 

841457 

4.49 

158543 

14 

47 

756236 

3 

o3 

9.45.0 

1-46 

84.726 

4-49 

158274 

i3 

48 

756418 

3 

o3 

914422 

1-46 

841996 

4-49 

i58oo4 

12 

49 

756600 

3 

o3 

914334 

1-46 

842266 

4-49 

157734 

11 

Do 

756782 

3 

02 

914246 

1-47 

842535 

4.49 

157465 

10 

5i 

9-756963 

3 

02 

9-9.4158 

1-47 

9-842805 

4-49 

10-157.95 

? 

52 

737144 

3 

02 

914070 

1-47 

843074 

4-49 

156926 

53 

757326 

3 

02 

9.3982 

1-47 

843343 

4-49 

156657 

7 

54 

757507 

3 

02 

913894 

1-47 

843612 

4-49 

156888 

6 

55 

757688 

3 

01 

9.3806 

1-47 

843882 

4-48 

i56.i8 

5 

56 

757869 

3 

01 

9.3718 

1-47 

844i5i 

4-48 

155849 

4 

u 

758o5o 

3 

01 

9i363o 

1-47 

844420 

4-48 

i555So 

3 

758230 

3 

01 

9.3541 

1-47 

844689 
844958 

4-48 

i553ii 

3 

59 

758411 

3 

01 

9.3453 

1-47 

4.48 

i55o42 

1 

60 

758591 

3.01 

913365 

1-47 

845227 

4-48 

154773  ' 

0 

Cosine 

D. 

Sine 

D. 

Cotang. 

D. 

Taiog. 

^J 

(55  DEGREES.) 


SINES   AND   TANGENTS.      (35    DEGREES.) 


58 


M. 

Sine 

D. 

Cosine  j  D. 

Tun,-. 

J). 

Cutani:. 

0 

9-758591 

3-01 

9-913365   1 

47 

9-845227 

4-48 

10- 154773  '  60 

I 

758772 

3 

00 

913276   I 

•47 

845496 

4-48 

1 54504  •   59  1 
1 54236  ;  58  ! 

2 

7D89-J2 

3 

00 

913187   I 

48 

845764 

4-48 

3 

759132 

3 

00 

913099   I 

48 

846033 

4-48 

153967 

57  i 

4 

759312 

3 

00 

9i3oio   I 

48 

846302 

4.48 

153698 

56  1 

5 

739492 

3 

00 

912922   I 

48 

846570 

4-47 

1 53430 

55 

6 

759672 

2 

99 

912833   I 

48 

846839 

4-47 

i53i6i 

54  ! 

I 

759852 

2 

99 

912744   I 

43 

847107 

4-47 

02893 

53  i 

76oo3i 

2 

99 

912655   I 

48 

847376 

4-47 

1 5262 i 

52  1 

9 

7602 1 1 

2 

99 

912566   I 

43 

847644 

4-47 

152356 

5i  1 

10 

760390 

2 

99 

912477   I 

48 

8479 '3 

4-47 

152087 

5o 

II 

9-760569 

2 

98 

9-912388   I 

48 

9-848181 

4-47 

io-i5i8i9 

^? 

12 

760748 

2 

98 

912299   I 

49 

848449 

4-47 

i5i55i 

i3 

760927 

2 

98 

912210   I 

49 

848717 

4-47 

i5i283 

47 

14 

761106 

2 

98 

912121    I 

49 

848986 

4-47 

i5ioi4 

46 

i5 

761285 

2 

98 

9i2o3i    I 

49 

849254 

4-47 

1 50746 

45 

i6 

761464 

2 

98 

911942   I 

49 

849322 

4-47 
4-46 

1 50478 

44 

n 

761642 

2 

97 

911853   I 

49 

849790 

l50210 

43 

i8 

761821 

2 

97 

91 1763   I 

49 

85oo58 

4-46 

149942 

42 

19 

761999 

2 

97 

911674   I 

49 

85o325 

4-46 

149675 

41 

20 

762177 

2 

97 

9ii584   I 

49 

85o593 

4.46 

149407 

40 

21 

g-762356 

2 

97 

9-911495   1 

49 

9-85o86i 

4-46 

10-149139 

39 
38 

"22 

762534 

2 

96 

911405   I 

49 

851129 

4.46 

148871 

23 

762712 

2 

96 

9ii3i5   I 

5o 

85i396 

4-46 

148604 

ll 

24 

762889 

2 

96 

911226   I 

5o 

85 1 664 

4-46 

148336 

25 

763067 

2 

96 

911136   I 

5o 

85 1 93 1 

4-46 

148069 

35 

26 

763245 

2 

96 

911046   I 

5o 

852.99 

4.46 

147801 

34 

27 

763422 

2 

96 

910906   I 

5o 

852466 

4.46 

147534 

33 

28 

763600 

2 

95 

9iot''66   I 

5o 

852733 

4-45 

147267 

32 

29 

763777 

2 

95 

910776   I 

5o 

853001 

4-45 

146999 
146732 

3i 

3o 

763954 

2 

95 

910686   I 

5o 

853268 

4-45 

So 

3i 

9-764131 

2 

95 

9-910596   I 

5o 

9-853535 

4-45 

10.146465 

ll 

32 

764308 

2 

95 

9io5o6   I 

5o 

853802 

4-45 

146198 
145931 

33 

764485 

2 

94 

9 1 o4 1 5   I 

5o 

854069 
854336 

4-45 

27 

34 

.  764662 

2 

94 

9io325   I 

5i 

4-45 

145664 

26 

35 

764838 

2 

94 

910235   I 

5i 

854603 

4-45 

145397 

25 

36 

765oi5 

2 

94 

910144   I 

5i 

854870 

4-45 

i45i3o 

24 

ll 

765 1 91 

2 

94 

910054  1  I 

5i 

855i37 

4-45 

144863 

23  • 

765367 

2 

94 

909963  1  I 

5i 

855404 

4-45 

144596 

22 

39 

765544 

2 

93 

909873   I 

5i 

855671 

4-44 

144329 

21 

4o 

765720 

2 

93 

909782   I 

5i 

855938 

4-44 

144062 

20 

4i 

9-765896 

2 

93 

9-909691    I 

5i 

9-856204 

4-44 

10.143796 

\l 

42 

766072 

2 

93 

909601    I 

5i 

856471 

4-44 

143520 
143263 

43 

766247 

2 

93 

909510   I 

5i 

856737 

4-44 

\l 

44 

766423 

2 

93 

909419   I 

5i 

857004 

4-44 

142996 

45 

766598 

2 

92 

909328   I 

52 

857270 

4-44 

142730 

i5 

46 

766774 

2 

92 

909237   I 

52 

857537 

4-44 

142463 

14 

47 

766949 

2 

92 

909146   I 

52 

857803 

4-44 

142197 

i3 

48 

767124 

2 

92 

909035   I 
908964   I 

52 

858069 

4-44 

141931 

12 

49 

767300 

2 

92 

52 

858336 

4-44 

141664 

II 

5o 

767475 

2 

91 

908873   I 

52 

858602 

4-43 

141398 

10 

5i 

9.767649 

2 

91 

9-908781    I 

52 

9-858868 

4-43 

io-i4n32 

? 

52 

767324 

2 

91 

908690   I 

52 

859134 

4-43 

140866 

53 

768173 

2 

91 

908599   I 

52 

859400 

4-43 

140600 

I 

54 

2 

91 

908507   I 

52 

859666 

4-43 

i4o334 

55 

768348 

2 

90 

908416   I 

53 

859932 

4-43 

140068 

5 

56 

768522 

2 

90 

908324   I 

53 

860198 

4 --43 

139802 

4 

III 

768697 

2 

90 

908233   I 

53 

860464 

4-43 

139536 

3 

768871 

2 

90 

908141    I 

53 

860730 

4-43 

139270 

a  \ 

59 

769045 

2 

90 

908049   I 

53 

860995 

4-43 

139005 
13^739 

I  1 

,  6c 

769219 

2-90 

907958   I 

.53 

861261 

4-43 

0  : 

i 

Cosine 

D. 

Sine     ] 

). 



Cotanar. 

D. 

Tunsr. 

(54    DEGREES.) 


5-1 


(36    DEGREES.)     A  TABLE   OF   LOGARITHMIC 


M. 

;   Siue 

1    "■ 

'  Cosine 

D. 

Tan?. 

D. 

Cotang. 

0 

!  9-769219 

IX. 

9-90-'958 
907866 

1.53 

9-861261 

4-43 

10-138739 

60 

I 

i    769393 

:  1.53 

8t.i527 

4-43 

;     138473 

lt\ 

2 

769566 

2.89 

907774  i  1-53 

861792 
862058 

;  4-42 

138208 

3 

769740 

2.89 

907682 

1  1-53 

!  4-42 

137942 

57 

4 

i    769913 

2.89 

907590 

1-53 

862323 

1  4-42 

137677 

56 

5 

770087 

2.89 

907498 

1-53 

862589 

4.42 

137411 

55 

6 

1    770260 

2-88 

907406 

1-53 

862854 

4-42 

137146 

54 

7 

770433 

2-88 

907314 

1.54 

863II9 
863385 

4.42 

1 36881 

53 

8 

!     770606 

2-88 

907222 

1.54 

4.42 

I366i5 

52 

1  9 

'     770779 

2-88 

907129 

1-54 

863650 

!  4-42 

136350 

5i 

10 

770952 

2-88 

907037 

1-54 

863915 

4-42 

i36o85 

5o 

11 

,  9-77II25 

2.88 

9-906945 
906852 

1.54 

9-864180 

4-42 

10-135820 

48 

12 

771298 

2.87 

1-54 

864445 

4-42 

135555 

i3 

i   771470 

2.87 

906760 

1.54 

864710 

4-42 

135290 

47 

14 

!   771643 

2.87 

906667 

1.54 

864975 

4-41 

135025 

46 

i5 

!     771S15 

2.87 

906575 

1-54 

865240 

4-41 

134760 

45 

i6 

771987 

2.87 

906482 

1-54 

8655o5 

4-41 

134493 

44 

\l 

772139 

2.87 

906389 

1-55 

865770 

4-41 

i3423o 

43 

'   77233f 

2-86 

906296 

1-55 

866035 

4-41 

133965 

42  ! 

19 

772303 

2-86 

906204 

1-55 

866300 

4-41 

i33-'oo 

41 

20 

772675 

2-86 

9061 1 1 

I  -55 

866564 

4-41 

133436 

40 

21 

9-772847 

2-86 

9-906018 

-  1-55 

9-866S29 

4-41 

10-133171 

39 

22 

773018 

2-86 

905925 

1-55 

867094 

4-41 

132906 

38 

23 

773190 

2-86 

9o5'^32 

1-55 

867358 

4-41 

132642 

37 

24 

773361 

2-85 

905739 

I  -55 

867623 

4-4i 

132377 

36 

25 

773533 

.  2-85 

903643 

1-55 

867887 

4-41 

132113 

35 

26 

773704 

2-85 

905552 

1-55 

8681 52 

4-40 

131848 

34 

27 

773875 

2-85 

905439 

1-55 

868416 

4-4o 

i3i5«4 

33 

28 

774046 

2-85 

905366 

1-56 

86S6S0 

4.40 

i3i32o 

32 

^9 

774217 

2-85 

905272 

1-56 

86S945 

4-40 

i3io55 

3i 

3o 

774388 

2-84 

905179 

1-56 

869209 

4-40 

130794 

3o 

3i 

9-774558 

2.84 

9-9o5o85 

1-56 

9-869473 

4.40 

io.i3o527 

l^ 

32 

774729 

2.84 

934992 
904898 

1-56 

869737 

4.40 

1 30263 

33 

774899 

2-84 

1-56 

870001 

4.40 

1 29999 

27 

34 

773070 

2-84 

904804 

1-56 

870265 

4.40 

1 29733 

26 

35 

775240 

2.84 

904711 

1-56 

870529 

4-40 

129471 

25 

36 

775410 

2-83 

904617 

1-56 

870793 

4-40 

129207 

24 

P 

775580 

2-83 

904523 

1-56 

871037 

4.40 

12S943 

23 

775750 

2-83 

904429 

1.57  : 

871321 

4.40 

12R6-9 

22  I 

39 

773920 

2-83 

904333 

1.57 

871535 

4.40 

128415 

21 

40 

776090 

2-83 

904241 

1.57 

871849 

4-39 

i28i5i 

20 

4i 

9-776259 

2-83 

9-904147 

1.57 

9.872112 

4-39 

10-127888 

;? 

42 

VXi 

2.82 

904053 

1-5-  , 

872376 

4-39 

127624 

43 

2.82 

903959 

1-5-  , 

872640 

4-39 

127360 

'7 

44 

776768 

2.82 

903864 

1-57 

872903 

4-39 

127097 

16 

45 

776937 

2.82 

903770 

1-57 

873167 

4-39 

I26S33 

i5 

46 

777106 

2.82 

903676 

1-57 

873430 

4-39 

126370 

i4 

^l 

777273 

2.81 

933581 

1.57 

873694 

4-39 

1263 06 

i3 

48 

777444 

2.81 

9034S7 

1-57 

873937 

4-39 

126043 

!2 

49 

777613 

2-8l 

903392 

1-58 

874220 

4-39 

123780 

11 

5o 

777781 

2-8l 

903298 

1-58  , 

874484 

4-3? 

I255i6 

10 

5i 

9-777950 

2.81 

9-9o32o3 

1-58  ' 

9-874747 

4-39 

10.125253 

t 

52 

7781 19 

2.81 

903108 

1-58  j 

875010 

4.39 

124990 

53 

778287 

2 -80 

9o3oi4 

1.58 

873273 

4-38 

124727 

- 

54 

77B455 

2.80 

902919 
902824 

1-58  1 

875536 

4-38 

124464 

6 

ii 

778624 

2-80 

1.58 

875800 

4-38 

124200 

5  1 

56 

778792 

2.80 

902729 

1.58  ! 

876063 

4-38 

123937 

4  1 

ll 

778960 

2-8o 

902634 

1-58  • 

876326 

4-38 

1 23674 

3 

779128 

2-8o 

902539 

1-59 

876589 

4-38 

1234.1 

2  1 

59 

779295 

2-79  : 

902444 

1.59 

876831 

4-38 

i23i4q 

I  i 

6o 

779463 

2-79  1 

902349 

1-59 

877114 

4-38  I 

122886 

0  ! 

1 

Cosine 

I). 

>^iiie 

D. 

Tan?. 

D. 

CotMntr.  '■ 

IHTJ 

(53  DEGREES.) 


SINES  AND   TANGENTS.      (37    DEGREESJ 


55 


M. 

Sine 

D. 

Cosine 

1   ^" 

1   Tang. 

D. 

Cotaiicr. 

o 

9-779463 

2-79 

9-902849 
902253 

1.59 

9-877114 

4-38 

10.122886  60 

I 

779681 

2 

79 

1.59 

877377 

4.38 

122628 

5? 

2 

77979'^ 

2 

79 

902(58 

1.59 

877640 

4-88 

122860 

3 

7799^6 

2 

79 

902068 

1.59 

877903 

4.38 

122097  57 

4 

780133 

2 

79 

901967 

1.59 

878.65 

4-38 

12.835   56 

5 

780300 

2 

7^ 

901872 

1.59 

878428 

4-38 

12.572   55 

6 

780467 

2 

"^5 

901776 

1.59 

878691 

4.38 

i2.3og 

54 

7 

780634 

2 

78 

901681 

I. 39 

878953 

4-87 

12.047 

53 

8 

780801 

2 

78 

901585 

1.59 

879216 

4-37 

120784 

52 

9 

780968 

2 

78 

901490 

1.59 

879478 

4-37 

120022   5l 

10 

781 1 34 

2 

78 

901894 

1.60 

879741 

4-37 

120259   5o 

II 

9-78i3oi 

2 

77 

9-901298 

1.60 

9.880008 

4.37 

10.119997   49 

12 

781 46S 

2 

77 

901202 

1.60 

880265 

4-37 

119735 

48 

i3 

781634 

■  2 

77 

90[io6 

1.60 

880528 

4-87 

1 19472 

47 

14 

781800 

2 

77 

901010 

1.60 

880790 
88io52 

4-37 

I192IO 

1.8948 

46 

i5 

781966 

2 

77 

900914 

1.60 

4-37 

45 

i6 

782(82 

2 

77 

900818 

1-60 

88i3i4 

4-37 

1 1 8686 

44 

\l 

782298 

2 

76 

900722 

1.60 

88.576 

4-87 

118424 

43  ' 

7S2464 

2 

76 

900626 

1.60 

88.889 

4-37 

1.8.61 

42 

'9 

782680 

2 

76 

900529 

1.60 

882.01 

4-87 

117899 

41 

20 

782796 

2 

76 

900433 

1. 6. 

882868 

4-36 

117637 

40 

21 

9.782961 

2 

76 

9.900837 

1. 61 

9.882625 

4-86 

10.117875 

ll 

''I 

788.27 

2 

7^ 

900240 

1-61 

882887 

4-86 

117113 

23 

788292 

2 

^i 

900144 

1-61 

888.48 

4-86 

1.6852 

37 

24 

788453 

2 

75 

ES 

1. 61 

888410 

4.86 

1.6590 

36 

25 

788628 

2 

75 

1-61 

888672 

4-36 

1.6828 

35 

26 

788788 

2 

75 

1. 61 

888984 

4-36 

1 1 6066 

34 

27 

788953 

2 

n 

899757 

i-6i 

884196 

4-36 

ii58o4 

38 

28 

7841 18 

2 

75 

899660 

i-6i 

884437 

4-36 

1.5543 

82 

29 

784282 

2 

74 

899564 

J. 61 

884719 

4.36 

1.523. 

3i 

So 

784447 

2 

74 

899467 

1.62 

884980 

4.36 

Il5020 

3o 

3i 

9.784612 

2 

74 

9.899870 

1.62 

9.885242 

4-36 

10.114758 

ll 

32 

784776 

2 

74 

899278 

1.62 

8855o8 

4-36 

114407 
114235 

33 

784941 

2 

74 

899176 

1.62 

885765 

4-36 

27 

34 

785 1 o5 

2 

74 

899078 

1.62 

886026 

4-36 

118914 

26 

35 

785269 

2 

73 

89S9S1 

1.62 

886288 

4.36 

118712 

25 

36 

785488 

2 

73 

898S84 

1.62 

886549 

4-85 

11845. 

24 

ll 

785597 

2 

73 

898787 

1.62 

8868.0 

4-35 

1 1 3 1 90 

23 

785761 

2 

73 

898689 

1.62 

887072 

4-35 

1.2928 

22 

39 

785925 

2- 

73 

898592 

1.62 

887888 

4-35 

1 . 2667 

21 

40 

786089 

2 

73 

89S494 

1.63 

887594 

4-35 

1.2406 

20 

41 

9.786252 

2 

72 

9.898897 

1.63 

9.887855 

4-35 

10.112145 

19 

,42 

786416 

2 

72 

89H299 

1.63 

8881 .6 

4-35 

1..8S4 

18 

43 

786579 

2 

72 

898202 

1.63 

888877 

4-35 

.1.628 

17 

44 

786742 

2 

72 

898104 

1.63 

888689 

4-35 

11.36. 

.6 

45 

786906 

2 

72 

89S006 

1.68 

888900 

4-35 

1 1 . 1 00 

.5 

46 

787069 

2 

72 

897908 

1.63 

889.60 

4-35 

110840 

14 

47 

787282 

2 

71 

897810 

1.63 

88912. 

4-35 

110079 

.3 

48 

787805 
787557 

2- 

71 

897712 

1.68 

8896S2 

4.35 

1103.8 

12 

49 

2. 

71 

8976.4 

1.63 

889943 

4-35 

110057 

II 

5o 

787720 

2. 

71 

897516 

1.63 

890204 

4-34 

109796 

10 

5i 

9.787883 

2. 

71 

9-897418 

1.64 

9.890465 

4-34 

10.109535 

1 

52 

788045 

2. 

71 

897820 

1.64 

890725 

4-34 

109275 

53 

788208 

2- 

71 

897222 

1.64 

890986 

4-34 

1090.4 

7 

54 

788870 

2 

70 

897.28 

1.64 

891247 

4-34 

108753 

6 

55 

788582 

2. 

70 

897025 

1-64 

89.507 

4-34 

108498 
108282 

5 

56 

788694 
788856 

2. 

70 

896926 

1.64 

891 76S 

4-34 

4 

u 

2- 

70 

896828 

1.64 

892028 

4-34 

107972 

3 

789018 

2 

70 

896729 

1-64 

892289 

4-34 

107711 

2 

59 

789180 

2 

70 

896681 

1.64 

892549 

4-34 

1 0745 1 

t 

60 

789342 

2.69 

896532 

1.64 

892810 

4.34 

107.90 

0 
M. 

Cosine 

D. 

Sine 

D. 

Cotang. 

D. 

Tang. 

(52    DEGREES.) 


(38    DEGRFES.)     A  TABLE   OF   LOGARITHMIC 


M. 

;"   Sine 

D. 

Cosine 

D. 

j  Tung. 

D. 

Cotang. 

6o" 

0 

9-789342 

2.69 

!  9.896532 

1.64 

9-892810 

:  4-34 

10- 107190 

I 

789504 

:  2.69 

896433 

1.65 

893070 

t  4-34 

106930 

'& 

2 

789665 

2.69 

896335 

1.65 

893331 

4.34 

106669 

3 

789827 

!  2-69 

896236 

1.65 

8938?! 

4.34 

106409 

57 

4 

789988 

2.69 

896137 

1-65 

4.34 

106149 

56 

5 

790 '49 

:  2-68 

896038 

1-65 

894111 

4.34 

1   105889 

55 

6 

7903 10 

895939 

1-65 

894371 

4-34 

io562o 
105368 

54 

1 

!  790471 

j  2.68 

!    895840 

1-65 

894632 

4.33 

53 

8 

j   790632 

2-68 

895741 

1-65 

894892 

4-33 

io5io8 

52  I 

9 

1   790793 
790954 

!  2.68 

895641 

1.65 

895152 

4-33 

104848 

5i  1 

10 

i  2-68 

1 

895542 

1-65 

895412 

4-33 

104588 

5o 

II 

i  9'79iii5 

1  2-68 

9-895443 

1-66 

9-895672 

4-33 

10.104328 

49 

12 

791275 

2.6"T 

895343 

1.66 

895932 

4.33 

104068 

48 

i3 

791436 

2.67 

895244 

1.66 

896192 

i  4-33 

io38o8 

47 

14 

791596 

2.67 

895145 

1-66 

896452 

i  4-33 

103548 

46 

i5 

.   791737 

2.67 

895045 

1-66 

896712 

1  4-33 

103288 

45 

i6 

791917 

2.67 

894945 

1.66 

896971 

1  4-33 

io3o29 

44 

n 

792077 

2.67 

894846 

1-66 

897231 

!  4-33 

102769 

43 

i8 

792237 

2.66 

894746 

1-66 

897491 

1  4-33 

102309 

42 

19 

792397 

2-66 

894646 

1-66 

897731 

4.33 

102249 

41 

20 

792DD7 

2.66 

894546 

1.66 

898010 

4-33 

101990 

40 

21 

9.792716 

2-66 

9.894446 

1.67 

9-89^270 

4.33 

10-101730 

\l 

22 

792876 

2.66 

894346 

1.67 

898530 

4-33 

101470 

23 

793035 

2.66 

894246 

1.67 

898789 

4.33 

101211 

37 

24 

793195 
7933D4 

2.65 

894146 

1-67 

899049 

4-32 

100951 

36 

23 

2-65 

894046 

1-67 

899308 

4-32 

100692 
100432 

35 

26 

793314 

2-65 

893946 

1-67 

899563 

4-32 

34 

27 

793673 

2.65 

893846 

1-67 

899827 

4-32 

100173 

33 

28 

793832 

2.65 

893745 

1.67 

900086 

4-32 

099914 

32 

29 

793991 

2-65 

893645 

1-67 

900346 

4-32 

099654 

3i 

3o 

7941 5o 

2.64 

893544 

1-67 

900605 

4.32 

099395 

3o 

3i 

9 -794308 

2.64 

9-893444 

1-68 

9-900864 

4-32 

10-099136 

\l 

32 

794467 

2.64 

893343 

1-68 

9011 24 

4-32 

098876 

33 

794626 

2.64 

893243 

1-68 

901383 

4-32 

09%  1 7 

27 

34 

794784 

2.64 

893142 

1.68 

901642 

4.32 

098358 ( 

26 

35 

794942 

2.64 

893041 

1.68 

901901 

4-32 

098099 

25 

36 

795101 

2.64 

892940 

1-68 

902160 

4.32 

097840 

24 

ll 

795259 

2.63 

892^139 

1.68 

902419 

4-32 

097581 

23 

38 

795417 

2-63 

892739 

1-68 

902679 

4-32 

097321 

22 

39 

795575 

2.63 

892638 

1-68 

902938 

4-32 

097062 

21 

40 

795733 

2-63 

892536 

1-68 

903197 

4-3i 

096803 

20 

41 

9.795891 

2.63 

9.892435 

1-69 

9.903455 

4-3i 

10-096545 

\l 

42 

796049 

2.63 

892334 

1.69 

903714 

4-3i 

096286 

43 

796206 

2.63 

892233 

1.69 

903973 

4-3i 

096027 

n 

44 

796364 

2.62 

892132 

1-69 

904232 

4-3i 

095768 

16 

45 

796521 

2.62 

892030 

1.69 

904491 
904750 

4-3i 

095509 

i5 

46 

796679 

2.62 

891929 

1-69 

4.31 

095250 

14 

47 

796836 

2.62 

891827 
891726 

1.69 

9o5oo8 

4-3i 

094992 

i3 

48 

796993 

2.62 

1.69 

905267 

4-3i 

094733 

12 

49 

797 1 30 

2.61 

891624 

1.69 

905526 

4-3i 

094474  1 

11 

DO 

797307 

2-61 

891523 

1.70 

905784 

4-3i 

094216 

10 

5i 

9-797464 

2.61 

9-891421 

1-70 

9.906043 

4-3i 

10.093957 

I 

52 

797621 

2.61 

89.319 

1-70 

906302 

4-3i 

093698  1 

53 

797777 

2.61 

891217 

1-70 

9o656o 

4-3i 

093440  i 

7 

54 

797934 

2-61 

89III5 

1.70 

9068 IQ 

4-3i 

093181  1 

6 

li 

79S091 

2.61 

89IOI3 

1.70 

907077 

4-31 

092923  1 

5 

56 

798247 

2.61 

890809 

1.70 

907336 

4-3i 

092664  : 

4 

f7 

798403 

2.60 

1-70 

907594 

4-3i 

092406  ! 

3 

58 

798560 

2-60 

890707 

1.70 

907802 

4-3i 

092148 

2  ! 

59 

798716 

2-6o 

800605 

1.70 

90S111 

4-3o  i 

091889 

1  i 

6c 

798872  , 

2-60 

890503 

1-70 

908369 

4-3o 

091631 

0  1 

Cosine 

D.   ' 

Sine 

I). 

Cotang.  ' 

D. 

Tang.   M.  \ 

(51 

DEGRE 

ES.) 

SINES   AKD   TANGENTS.      (39    DEGREES.) 


57 


M. 
o 

Sine 

i   ^• 

,  Cosine 

D. 

Tang. 

D. 

.  Cotaug. 

9-79S872 

:  2-60 

I  9-890503 

1.70 

9.908369 

:  4-3o 

10.091631  j  60 

I 

799028 

\       2-60 

1   890400 

1. 71 

908628 

;  4-3o 

091372  1  59 

2 

,   799 '^4 

j  2-6o 

89029S 

1. 71 

908886 

i  4-3o 

091 1 14  I  58 

3 

i   •/99^-^9 

2-59 

890195 

1. 71 

909144 

;  4-3o 

090856  j  57 

4 

1   799493 

2-59 

890093 

i-7> 

909402 

j  4-3o 

090598   56 

5 

;   799^31 

2-59 

889990 

1. 71 

909660 

i  4-3o 

090340  i  55 

6 

799806 

2.59 

889888 

1. 71 

909918 

4-3o 

090082   54  1 

7 

;    799962 

2-59 

8H9785 

1. 71 

910177 

4-3o 

089823   53  ; 

8 

800117 

2.59 

.   8,S9b82 

1.71 

910435 

4-3o 

089565  !  52  ! 

9 

800272 

2.58 

889579 

1-71 

910693 

4-3o 

089307  !  5i  ! 

10 

800427 

2.58 

1   889477 

1. 71 

9109D1 

4-3o 

089049  j  DO 

u 

9.800582 

2-58 

i  9-889374 

1.72 

9.911209 

!  4-3o 

10.088791  1  49 

12 

800737 

2.58 

j   889271 

1-72 

91 1467 

1  4-3o 

08S533  !  48 

i3 

800892 

2-58 

889168 

1-72 

911724 

4-3o 

088276  i  47 

14 

801047 

2-58 

889064 

1.72 

9119S2 

4-3o 

088018 

46 

ID 

801201 

2.58 

88.S961 

1-72 

912240 

4-3o 

087760 

45 

i6 

8oi3d6 

2.57 

888858 

1-72 

912498 

4-3o 

087502 

44 

'7 

801D11 

2.57 

888755 

1.72 

912756 

4-3o 

087244 

43 

i8 

801 665 

2.57 

88S65I 

1.72 

9i3oi4 

4-29 

0869S6 

42 

'9 

801819 

2. 57 

88,s548 

1.72 

913271 

4-29 

086729 

41 

20 

801973 

2.57 

888444 

1.73 

913529 

4-29 

086471 

40 

21 

9.802128 

2.57 

9-888341 

1.73 

9-913787 

4-29 

10.086213 

ll 

22 

802282 

2.56 

88:^237 

1.73 

914044 

4-29 

085956 

23 

802436 

2.56 

888134 

1.73 

914302 

4-29 

085698 

37 

24 

802D89 

2-56 

888o3o 

1.73 

914560 

4-29 

085440 

36 

25 

802743 

2-56 

887926 

1.73 

914817 

4-29 

o85i83 

35 

26" 

802897 

2.56 

887822 

1.73 

915075 

4-29 

084925 

34 

27 

8o3oDo 

2.56 

887718 

1.73 

915332 

4-29 

084668 

33 

28 

803204 

2-56 

887614 

1.73 

915590 

4-29 

084410 

32 

l") 

803357 

2.55 

887510 

1.73 

9 '5847 

4-29 

084153 

3i 

3o 

8o35ii 

2.55 

887406 

1-74 

916104 

4-29 

083896 

3o 

3. 

9-803664 

2.55 

9.887302 

1-74 

9-916362 

4.29 

10.083638 

29 

32 

803817 

2.55 

88719^ 

1-74 

9 1 66 1 9 

4-29 

o833Si 

28 

33 

803970 

2.55 

887093 

1-74 

916877 

4-29 

o83i23 

27 

?4 

804123 

2.55 

886989 

1-74 

917134 

4-29 

082866 

26 

35 

804276 

2.54 

886885 

1-74 

917391 

4-29 

082609 

25 

36 

804428 

2-54 

886780 

1-74 

917648 

4-29 

082352 

24 

3-» 

804581 

2.54 

886676 

1-74 

917905 

4-29 

082095 

23 

38 

804734 

2.54 

886571 

1-74 

918163 

4-28 

081837 

22 

39 

804886 

2.54 

8864()6 

1-74 

918420 

4-28 

081 580 

21 

4o 

8o5o39 

2.54 

886362 

,.75 

918677 

4-28 

o8i323 

20 

4i 

9-805191 

2.54 

9.886257 

1.75 

9-918934 

4-28 

10.081066 

\l 

42 

805343 

2.53 

886152 

1.75 

919191 

4-28 

080809 

43 

805493 

2-53 

886047 

1.75 

919448 

4-28 

o8o552 

n 

44 

8o5647 

2.53 

885942 

1.75 

919705 

4-28 

080295 

16 

45 

805799 

2-53 

885837 

1.75 

919962 

4.28 

o8oo38 

ID 

46 

80.59)1 

2-53 

885732 

1-75 

920219 

4-28 

079781 

14 

47 

806 io3 

2-53 

885627 

I-7D 

920476 

4-28 

079524 

i3 

48 

806254 

2.53 

88552  2 

1.75 

920733 

4-28 

079267 

12 

49 

806406 

2-52 

885416 

1-75 

920990 

4-28 

079010 

II 

Do 

806557 

2-52 

8853 1 1 

1.7b 

921247 

4-28 

078753 

10 

5. 

9 • 806709 

2.52 

9.885205 

1.76 

9.921503 

4-28 

10.078497 

t 

52 

806860 

2.52 

885100 

1.76 

921760 

4-28 

078240 

53 

8070 1 1 

2.52 

884994 

1.76 ! 

922017 

4-28 

0779S3 

I 

54 

807163 

2.52 

884889 

1-76 

922274 

4-28 

077726 

55 

807314 

2.52 

884783 

1-76 

922530 

4-28 

077470 

5 

56 

807465 

2.5l 

884677 

1.76 

922187 

4.28 

077213 

4 

57 

807615 

2-5l 

884572 

1.76 

923044 

4-28 

076956 

3 

58 

807766 

2-5l 

884466 

1-76 

923300 

4-28 

076700 

2 

59 

807917 

2.5l 

884360  : 

1.76 ' 

923557  ! 

4.27 

076443 

I 

60 

808067 

2-5l 

884254 

'•77 

923813  1 

4-27 

076187  1  0 

Cosine 

D. 

Sine 

D. 

Cotantr.  1 

D. 

Tiuig.  i  M. 

(50    UEGUKES.) 


58 


(40    DEGREES.)     A  TABLE   OF   LOGARITHMIC 


0 

Sine 

,   D. 

'  Cosine 

D. 

Tun-. 

L>. 

i  Cotang.  j 

9.808067 

!  2-5i 

9.884254 

1-77 

,  9-9238i3 

4-27 

10-076187  1  60 

I 

808218 

;   2-51 

884148 

1-77 

j   924070 

1  4-27 

075930   59 
070673,!  58 

2 

8o8368 

1   2-5l 

'   884042 

1-77 

924327 

1  4-27 

3 

8oS5i9 

2.5o 

1   883936 

1-77 

1   92  4583 

1  4-27 

075417  \   57 
075160  I  56 

4 

808669 

200 

1   883829 

1-77 

!   924840 

i  4-27 

5 

808819 

2-5o 

1   883723 

'  1-77 

j   925096 

;  4-27 

074904  1  55 

6 

808969 

2.5o 

i   883617 

'  1-77 

;   925332 

4-27 

074648  i  54 

7 

8091 19 

2 -50 

!   883510 

1-77 

i   925609 

4-27 

074301  1  53 
074135  ]  52 

8 

809269 

'   2 -50 

1   883404 

1-77 

!   925865 

4-27 

9 

809419 

2-49 

!   883297 

1.78 

926122 

i  4-27 

073878 

5i 

10 

809569 

'  2.49 

883191 

;  1-78 

1   926378 

i  4-27 

073622 

50  1 

u 

9-8o97i8 

2-49 

1  9 -883084 

1.78  1  9-926634 

i  4.27 

10-073366 

t? 

12 

809S68 

2-49 

882977 

,  1-78 

;  926890 

4-27 

073110 

i3 

8ioot7 

2-49 

882871 

1.78 

927147 

4-27 

072803 

47  1 

14 

810167 

2-49 

882764 

!  1-78 

1  927403 

4-27 

072097  i  46  1 

i5 

8io3i6 

2-48 

.882657 

1  1-78 

,  927609 

!   927915 

4-27 

072341  i  45 

i6 

810465 

2.48 

882550 

1  1-78 

4-27 

072085 

44 

n 

810614 

2-48 

882443 

1-78 

1  928I7I 

4-27 

071829 

43 

i8 

810763 

2-48 

882336 

1-79 

1  928427 

4-27 

071573 

42 

19 

810912 

2-48 

882229 

1-79 

928683 

4-27 

071317 

41 

20 

8iio6j 

2.48 

882121 

1-79 

928940 

4-27 

071060 

40 

21 

9-811210 

2-48 

9-882014 

1-79 

1  9.929196 

4-27 

10-070804 

ll 

22 

8ii358 

2-47 

881907 

1.79 

1  929402 

4.27 

070548 

23 

8ii5o7 

2-47 

88 1 799 

1.79 

1  92970S 

4-27 

070292 

11 

24 

8ii655 

2-47 

881692 

1-79 

j  929964 

4-26 

070036 

25 

81 1804 

2-47 

88 1 584 

1-79 

1   930220 

4-26 

069780 

35 

26 

811952 

2-47 

881477 

1.79 

1  930475 

4-26 

069525 

34 

27 

812100 

2-47 

881369 

1-79 

!  930731 

4-26 

069269 

33 

28 

812248 

2.47 

8S1261 

1-80 

930987 

4-26 

069013 

32 

29 

812396 

2-46 

88 1 1 53 

1.80 

931243 

4-26 

068757 

3i 

So 

812044 

2-46 

881046 

1.80 

931499 

4-26' 

06856 1 

30 

3i 

9-812692 

2.46 

9-880938 

1.80 

9.931755 

4-26 

10.068245 

ll 

32 

812840 

2.46 

8So83o 

I -80 

932010 

4.26 

067990 

33 

812988 

2.46 

880722 

1-80 

932266 

4-26 

067734 

27 

34 

8i3i35 

2  ■  46 

880613 

1-80 

932522 

4.26 

067478 

26 

35 

8i32S3 

2.46 

8So5o5 

1.80 

932778 

4-26 

067222 

25 

36 

8i343o 

2-45 

8S0397 

1-80 

933033 

4.26 

066967 

24 

37 
38 

813578 

2.45 

88o2:s9 

i-8i 

933289 

4.26 

0667 ' I 

23 

813725 

2.45 

880180 

i-8i 

933545 

4-26 

066455 

22 

39 

813872 

2.45 

880072 

i-8i 

933800 

4-26 

066200 

21 

40 

814019 

2.45 

879963 

1-81 

934056 

4-26 

065944 

20 

41 

9.814166 

2.45 

9.879855 

1.81 

9.934311 

4-26 

10.060689 

;i 

42 

8i43i3 

2-45 

879746 

i-8i 

934567 

4-26 

060433 

43 

814460 

2-44 

879637 

1. 81 

934823 

4-26 

060177 

17 

44 

814607 

2-44 

879529 

i-8i 

935078 

4-26 

064922 

16 

45 

814753 

2-44 

879420 

r.8i 

935333 

4-26 

064667 

i5 

46 

814900 

2-44 

879311 

1-81 

935589 

4-26 

064411 

14 

47 

8i5o46 

2-44 

879202 

1-82 

930844 

4-26 

064106 

i3 

48 

815193 

2-44 

879093 

1-82 

936100 

4-26 

063900 

12 

49 

815339 

2-44 

8780S4 
878875 

1.82 

936355 

4-26 

o63640 

11 

5o 

8 1 5485 

2.43 

1-82 

936610 

4-26 

063390 

10 

5i 

9-8i563i 

2.43 

9-878766 

1-82 

9-936866 

4-20 

10-063 1 34 

I 

52 

815778 

2.43 

878656 

1.82 

937121 

4-20   , 

062^^79 

53 

815924 

2.43 

878547 

1-82 

937316 

4-25  i 

062624 

7 

54 

816069 

2-43 

878438  1 

1-82 

937632 

4-25 

06236S  ! 

6 

55 

816215 

2.43 

878328 

1-82 

937887 

4-25 

062113  '■ 

5 

56 

8i636i 

2.43 

878219 

1.83 

938142 

4-25 

061808  : 

4 

57 

8i65o7 

2-42 

878109  ! 

1-83 

938398 

4-25 

061602  : 

3 

58 

8i6652 

2.42 

877999 

1.83 

938653 

4-25 

061347 

2 

59 

816798 

2-42 

877S90 

I.  S3 

93^^908 

4-25  1 

061092 

I 

60 

816943 

2  •  42 

8777H0 

1.83 

939163 

4-20 

060837 

0 

Cosine 

I). 

Sine 

D. 

CotunjT.  1 

I). 

Tunff.   ;  M.  1 

(49 

DEGRI 

:es.) 

SINES   AND   TANGENTS.      (41    DEGREES.) 


59 


M. 

Sine 

D. 

Co>iiie    D. 

_Z:^^_ 

U. 

Cotang. 

1 

1 

0 

9.816943 

2-42 

9.877780   I 

83 

9.939163 

4-25 

10-060837 

60 

1 

8170:^3 

2 

42 

877670   1 

83 

939418 

4 

25 

o6o582 

11 

2 

817233 

2 

42 

877360  :  I 

83 

939673 

4 

25 

o6o327 

3 

817379 

2 

42 

877450  ,  I 

83 

939923 

4 

25 

060072 

37 

4 

817324 

2 

41 

877340  1  I 

83 

940183 

4 

25 

0598.7 

56 

5 

817668 

2 

41 

877230   I 

84 

940438 

4 

23 

059362 

55 

6 

817813 

2 

41 

877120  :   I 

84 

940694 

4 

25 

059306 

54 

7 

817958 

2 

41 

877010  :   I 

84 

940949 

4 

25 

059051 

53 

8 

8i8io3 

2 

41 

876899  i   I 

84 

941204 

4 

25 

058796 

52 

9 

818247 

2 

41 

876789  i   I 

84 

941458 

4 

25 

058542 

31 

10 

818392 

2 

41 

876678  i   I 

84 

941714 

4 

25 

058286 

5o 

II 

9.818536 

2 

40 

9.876568  i   I 

84 

9.941968 

4 

25 

io.o58o32 

49 

12 

8 1868 1 

2 

40 

876457  i   I 

84 

942223 

4 

25 

037777 

48 

i3 

8i8:J25 

2 

40 

876347  1   I 

84 

942478 

4 

25 

057D22 

47 

i4 

818969 

2 

40 

876236  1   I 

85 

942733 

4 

25 

037267 

46 

i5 

819113 

2 

40 

676125  !   I 

85 

9429^8 

4 

25 

057012 

45 

i6 

819257 

2 

40 

876014  1   I 

tl 

943243 

4 

25 

036757 

44 

17 

819401 

2 

40 

875904  :   I 

943498 

4 

25 

o565o2 

43 

iS 

819045 

2 

39 

875793  i  I 

85 

943752 

4 

25 

056248 

42 

19 

819689 

2. 

39 

875682  I   I 

85 

944007 

4 

25 

055993 

41 

20 

819832 

2 

39 

875571  .   I 

85 

944262 

4 

35 

055738 

40 

21 

0-819976 

2 

39 

0-875459  .   I 

85 

9-944517 

4 

25 

10-055483 

39 

22 

820120 

2 

39 

875348     I 

85 

9 '•4771 

4 

24 

055229 

38 

23 

820263 

2 

39 

875237     I 

85 

945026 

4 

24 

054974 

37 

24 

820406 

2 

39 

875126     I 

86 

943281 

4 

24 

054719 

36 

2J 

82o33o 

2 

38 

875014  1   I 

86 

945535 

4 

24 

054463 

35 

26 

820693 

2 

38 

874903  1   I 

86 

945790 

4 

24 

054210 

34 

27 

820836 

2 

38 

874791  1  I 

86 

946045 

4 

24 

053955 

33 

28 

820979 

2 

38 

874680  i   I 

86 

946299 

4 

24 

053701 

32 

29 

821122 

2 

38 

874568  1   I 

86 

946554 

4 

24 

053446 

3i 

3o 

821265 

2 

38 

874456  ;  I 

86 

946808 

4 

24 

053.92 

3o 

3i 

9-821407 

2 

33 

9.874344  t  I 

86 

9-947063 

4 

24 

10-052937 

11 

32 

82i55o 

2 

38 

874232  !  I 

87 

947318 

4 

24 

052682 

33 

821693 

2 

37 

874121  ;  I 

87 

947572 

4 

24 

052428 

27 

34 

821835 

2 

37 

874009  1  I 

87 

947826 

4 

24 

052174 

26 

33 

821977 

2 

37 

873896  1  I 

87 

94S081 

4 

24 

05.919 

25 

36 

822120 

2 

37 

873784  :   I 

87 

948336 

4 

24 

05.664 

24 

37 

822262 

2 

37 

873672  1  I 

87 

94S590 

4 

24 

o5i4io 

23 

38 

822404 

2 

37 

873560  1  I 

87 

948844 

4 

24 

o5ii56 

22 

39 

822546 

2 

37 

873448  i  I 

87 

949099 
949353 

4 

24 

050901 

21 

1  4o 

822688 

2 

36 

873335  1  I 

87 

4 

24 

050647 

20 

41 

9.822830 

2 

36 

9.873223  ,  I 

87 

9-949607 

4 

24 

io-o5o393 
o5oi38 

\l 

42 

822972 

2 

36 

8731.0  !   I 

88 

949862 

4 

24 

43 

823114 

2 

36 

872993  i   I 

88 

930116 

4 

24 

049884  1  17  1 

44 

823235 

2 

36 

872885  j   I 

88 

950370 

4 

24 

049630 

•6 

4J 

82  5397 

2 

36 

872772     I 

88 

930623 

4 

24 

049373 

i5 

46 

823339 

2 

36 

872659     I 

88 

930879 

4 

24 

049.2. 

14 

47 

823680 

2 

35 

872547     I 

88 

93 11 33 

4 

24 

048867 

i3 

48 

823821 

2 

35 

872434     I 

83 

95 1 388 

4 

24 

0485.2 

12 

49 

823963 

2 

35 

872321      I 

83 

931642 

4 

24 

043353 

II 

DO 

824104 

2 

35 

872208     I 

83 

951896 

4 

24 

048104 

10 

5i 

9-824245 

2 

35 

9.872093     I 

89 

9.95200 

4 

24 

I0-047850 

t 

52 

824386 

2 

33 

871981     I 

89 

952405 

4 

24 

047595 

53 

824527 

2 

35 

87 1 808     I 

89 

$f^i 

4 

24 

047341 

7 

54 

824668 

2 

34 

871755     I 

89 

4 

24 

047087 

6 

55 

824S03 

2 

34 

871641  1   I 

89 

953.67 

4 

23 

046333 

5 

56 

824949 

2 

34 

871528  !  I 

89 

933421 

4 

23 

046379 

4 

57 

823090 

2 

34 

871414   I 

89 

933675 

4 

23 

046323  !  3  1 

58 

823230 

2 

•34 

871301   I 

89 

933929 

4 

23 

046071  ■  2 

59 

825371 

2 

34 

871 187   I 

89 

934.83 

4 

23 

0438.7    I 

60 

8255ii 

2-34 

871073   I 

90 

954437 

4-23 

043563   0 

L 

O'jsine 

1    D. 

'  Sine   1  1 

). 

Cotan^. 

D. 

Tanff.   M. 

(43    DEGREES.) 


60 


(42   DEGREES.)      A  TABLE   OF   LOGARITHMIC 


M. 
o 

Sine 

D. 

Cusine 

D- 

TaUiT. 

D. 

(Jotang. 

9-8255ii 

2-34 

9-871073 

1.90 

9-954437 

4.23 

10-045363 

60 

I 

825651 

2 

33 

870960 

1-90 

954691 

4-23 

045309   5q 
o45o55  58 

2 

'   820791 

2 

33 

870846 

1-90 

1   954945 

4-23 

3 

'•   825931 

2 

33 

870732 

1.90 

'   955200 

4-23 

044800  57 

4 

826071 

2 

33 

870618 

1-90 

955454 

4-23 

044546  56 

5 

8262 1 1 

2 

33 

870304 

1.90 

955707 

4-23 

044293   55 

6 

826351 

2 

33 

870390 

1-90 

955961 

4-23 

044039  54 

7 

826491 
826631 

2 

Zi 

870276 

1-90 

956215 

4-23 

043780   53 

8 

2 

33 

870161 

1-90 

'Xt 

4-23 

043531   52 

9 

826-70 

2 

32 

870047 

1-91 

4-23 

1    043277   31 

lO 

826910 

2 

32 

869933 

I-9I 

9D6977 

4-23 

o43o23   5o 

II 

9-827049 

2 

32 

9-869818 

1-91 

9-957231 

4-23 

10-042769   49 
o425i5   48 

12 

827180 
827328 

2 

32 

869704 

1. 91 

957485 

4-23 

i3 

2 

32 

869589 

1-91 

957739 

4-23 

042261   47 

14 

827467 

2 

32 

869474 

1-91 

957993 

4-23 

042007   46 

i5 

827606 

2 

32 

869360 

1-91 

958246 

4-23 

041754 

45 

i6 

827745 

2 

32 

869245 

1-91 

958500 

4-23 

o4i5oo 

44 

n 

827884 

2 

3i 

869130 

1-91 

95S754 

4-23 

041246 

43 

i8 

82B023 

2 

3i 

869015 
86S900 

1-92 

959008 

4-23 

040992 

42 

19 

828162 

2 

3i 

1-92 

939262 

4-23 

040738 

41 

20 

828301 

2 

3i 

868785 

1-92 

959516 

4-23 

040484 

40 

21 

9-828439 

2 

3i 

9-868670 

1-92 

9-959769 
960023 

4-23 

10  04023 1 

3I 

22 

828578 

2 

3i 

868555 

1-92 

4-23 

039977 

23 

828716 

2 

3i 

868440 

1-92 

960277 

4-23 

039723 

37 

24 

828855 

2 

3o 

868324 

1-92 

960531 

4-23 

039469 

36 

25 

82B993 

2 

3o 

868209 
868093 

1-92 

960784 

4-23 

039216 

35 

26 

829131 

2 

3o 

1-92 

961038 

4-23 

03:^962 

34 

27 

829269 

2 

3o 

867978 

1.93 

961291 

4-23 

038709 

33 

28 

829407 

2 

3o 

867862 

1.93 

961545 

4-23 

o3S45o 

32 

29 

829545 

2 

3o 

867747 

1.93 

961799 

4-23 

o3S2oi 

3i 

3o 

829683 

2 

3o 

867631 

1.93 

962052 

4-23 

037948 

3o 

3i 

9.829821 

2 

29 

9-8675i5 

1-93 

9.962306 

4-23 

10-037694 

It 

32 

829959 

2 

29 

^,\V^ 

1-93 

962560 

4-23 

037440 

33 

830097 

2 

29 

1-93 

962813 

4-23 

037187 
036933 

11 

34 

830234 

2 

29 

86-1161 

1-93 

963067 

4.23 

35 

83o372 

2 

29 

867051 

1-93 

963320 

4-23 

o3668o 

25 

36 

83o5o9 

2 

29 

866935 
866819 
866703 

1-94 

963574 

4-23 

036426 

24 

37 

83o646 

2 

29 

1-94 

963827 

4-23 

o36i73 

23 

38 

830784 

2 

29 

1-94 

964081 

4-23 

035919 

22 

39 

830921 

2 

28 

866586 

1-94 

964335 

4-23 

o3566o 

21 

40 

83io58 

2 

28 

866470 

1-94 

964588 

4-22 

o354i2 

20 

41 

9-831195 

2 

28 

9-866353 

1-94 

9-964842 

4-22 

io-o35i58 

;? 

42 

83i332 

2- 

28 

866237 

1-94 

965095 

4-22 

o349o5 

43 

831469 

2 

28 

866120 

1-94 

965349 

4-22 

o3465i 

17 

44 

83 1 606 

2 

28 

866004 

1-95 

965602 

4-22 

034398 

16 

45 

•  831742 

2 

28 

865887 

1-95 

965855 

4-22 

034145 

i5 

46 

831879 

2 

28 

865770 

1.95 

966105 

4-22 

033891 
033638 

14 

47 

8320ID 

2 

27 

865653 

1-95 

966362 

4-22 

i3 

48 

832152 

2 

27 

865536 

1-95 

966616 

4-22 

033384 

12 

U 

832288 
832425 

2 
2 

27 
27 

865419 
865302 

1.95 
1.93 

» 

4-22 
4-22 

o33i3i 

032877 

II 
10 

5i 

9-832561 

2 

27 

9-865185 

1.95 

9-967376 

4-22 

10-032624 

t 

52 

832697 

2 

27 

865o63 

1-95 

&^ 

4-22 

032371 

53 

832833 

2 

27 

864900 

1-95 

4-22 

032117 

7 

54 

832969 

2 

26 

864833 

1-96 

968136 

4-22 

o3i864 

6 

55 

833 loD 

2 

26 

864716 

1.96 

968389 

4-22 

o3i6ii 

5 

56 

833241 

2 

26 

864598 
864481 

1.96 

968643 

4-22 

o3i357 

4 

% 

833377 

2 

26 

1-96 

968896 

4-22 

o3iio4 

3 

833512 

2 

26 

864363 

1-96 

969149 

4-22 

o3oS5i 

2 

59 

833648 

2 

26 

864245 

1-96 

969403 

4-22 

o3o097 

I 

60 

833783 

2-26 

864127 

1.96 

969656 

4.22 

o3o344 

0 

Cosine 

D. 

Sine 

D. 

Cotanor.  i 

D. 

Tang.    M.  | 

(47  DEGREES.) 


SINES  AND   TANGENTS.      (43    DEGREES.) 


61 


M. 

Sine 

D. 

Cosine 

D. 

Tung. 

D. 

Cotang. 

0 

9-833783  1 

2-26 

9-864127  j 

1-96 

9-969656 

4-22 

io-o3o344 

60 

333919 

2-25 

864010 

1.96 

969909 

4-22 

030091   59 

3 

834034 

2-25 

863892 

1-97 

970162 

4-22 

029838  58 

3 

834189 

2-25 

863774 

1-97 

9704x6 

4-22 

029584  '  57 

4 

83432D  , 

2-25 

863656 

1-97 

970669 

4-22 

029331   56 

5 

834460 

2-25 

863538 

1-97 

970922 

4-22 

029078  55 

6 

834595 

2-25 

■  863419 

1-97 

971175 

4.22 

028825 

54 

I 

834730 

2-25 

863301  ; 

1-97 

971429 

4.22 

028571 

53 

834865 

2-25 

863 1 83 

1-97 

97x682 

4-22 

0283 1 8 

52 

9 

834999 

2-24 

863o64  ' 

1.97 

971935 

4-22 

028065 

5i 

10 

835i34 

2-24 

862946 

1.98 

972188 

4-22 

027812 

5o 

11 

9.835269 

2-24 

9-862827  1 

1-98 

9-972441 

4-22 

10-027559 

% 

12 

835403 

2-24 

862709  1 

1 .98 

972694 

4-22 

027306 

i3 

835538 

2-24 

862590 

1-98 

972948 

4-22 

027052 

47 

U 

835672 

2-24 

862471  : 

1.9S 

973201 

4-22 

026799 

46 

i5 

835807 

2-24 

862353 

1-98 

973454 

4-22 

026546 

45 

i6 

835941 

2-24 

862234 

1.98 

973707 

4-22 

025293 

44 

\l 

836075 

2-23 

862X15  1 

1-98 

973960 

4-22 

026040 

43 

836209 
836343 

2-23 

861996 

1.98 

9742x3 

4-22 

025787 

42 

19 

2-23 

861877 

1.98 

974466 

4-22 

025534 

41 

20 

836477 

2-23 

861758 

1.99 

974719 

4-22 

025281 

40 

21 

9.8366n 

2-23 

9-86x638 

1-99 

9-974973 

4-22 

10-025027 

^ 

22 

836745 

2-23 

86x5x9 

1.99 

975226 

4-22 

024774 

23 

836878 

2-23 

861400 

1-99 

975479 

4.22 

024521 

37 

24 

837012 

2-22 

86x280 

1-99 

975732 

4-22 

024268 

36 

25 

837146 

2-22 

861 x6x 

1.99 

■975985 

4-22 

0240x5 

35 

26 

837279 

2-22 

86 1 04 1 

1.99 

976238 

4-22 

023762 

34 

27 

837412 

2-22 

860922 
860802 

1.99 

976491 

4-22 

023509 
023256 

33 

28 

837546 

2-22 

1-99 

976744 

4-22 

32 

29 

837679 

2-22 

860682 

2.00 

976997 
9772DO 

4-22 

023oo3 

3i 

3o 

837812 

2-22 

86o562 

2-00 

4-22 

022750 

3o 

3i 

9 -537945 

2-22 

9-860442 

2-00 

9-9775o3 

4.22 

10.022497 

^i 

32 

838078 

2-21 

86o32  2 

2-00 

977756 

4-22 

022244 

33 

83S2II 

2-21 

860202 

2-00 

978009 

4-22 

021991 

27 

34 

838344 

2-21 

860082 

2-00 

978262 

4.22 

021738 

26 

35 

838477 

2-21 

859962 

2-00 

9785,5 

4.22 

021485 

25 

36 

8386x0 

2-21 

859842 

2-00 

978768 

4.22 

021232 

'^ 

ll 

838742 

2-21 

85972 X 

2-OI 

979021 

4-22 

020979 

23 

838875 

2-21 

85960 X 

2-OX 

979274 

4.22 

020726 

22 

39 

839007 

2-21 

859480 

2-01 

979527 

4.22 

020473 

21 

40 

839140 

2-20 

859360 

2-01 

979780 

4.22 

020220 

;o 

41 

9-839272 

2-20 

9.859239 

2.01 

9-980033 

4-22 

10-019967 

\i 

42 

839404 

2-20 

8591 iQ 

2.01 

980286 

4-22 

OI97I4 

43 

839536 

2-20 

85M99B 

2.01 

980538 

4-22 

019462 

17 

44 

839668 

2-20 

85S877 

2.01 

9S0791 

4-21 

019209 

16 

45 

839800 

2-20 

858756 

2.02 

981044 

4.21 

018956 

i5 

46 

839932 

2-20 

858635 

2.02 

$?,Vo 

4.21 

018703 

'i 

47 

840064 

2-19 

8585x4 

2-02 

4.21 

018450 

i3 

48 

840196 

2-19 

858393 

2.02 

'   9S1803 

4.21 

OI8I97 

12 

49 

840328 

2-19 

858272 

2-02 

1   982056 

4.21 

017944 

11 

5o 

840459 

2-19 

858i5i 

2.02 

1   982309 

4-21 

017691 

10 

5i 

9-840591 

2-19 

9-858029 
857908 

2.02 

9-982562 

4-21 

10.017438 

I 

52 

840722 

'   2-19 

2.02 

982814 

4-21 

017186 

53 

840854 

2-19 

857786 

2-02 

9^^3067 

4-21 

016933 

I 

54 

840985 

I'M 

857665 

2 -03 

9S3320 

4-21 

016680 

6 

55 

841 1 16 

857543 

2-o3 

983573 

4-21 

0x6427 

5 

56 

841247 
841378 

;   2.18 

857422 

2-03 

983826 

!  4-21 

016174 

4 

57 

2.18 

857300 

2-o3 

;   9*^4079 

4-21 

015921 

3 

58 

841509 

'   2.18 

857x78 

2-o3 

9843 3 X 

4-V!l 

015669  -     1 

59 

841640 

2-18 

857056 

2-o3 

9^45^4 

4-21 

015416 

I 

to 

841771 

;  2-18 

856934 

2.03 

984837 

4-21 

oi5i63 

0 
M. 

1  Cosine 

1   D. 

1   Sine 

D. 

Cotanar. 

i   D. 

'  Tang. 

(46    DEGREES.) 


C2 


(44   DEGREES.)     A  TABLE   OF   LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotang.  1 

0 

9-841771 

2.18 

9.856934 

2-03 

9-984837 

4-21 

io-oi5i63 

60 

I 

841902 

2.18 

8568x2 

2-03 

985090 

4 

21 

014910 

u 

2 

842033 

2 

18 

856690 

2-04 

9S5343 

4 

21 

014657 

3 

842163 

2 

17 

856568 

2-04 

985596 

4 

21 

014404 

57 

4 

842294 

2 

17 

856446 

2-04 

985848 

4 

21 

014132 

56 

5 

842424 

2 

17 

856323 

2-04 

986101 

4 

2! 

013899 

55 

6 

842555 

2 

•7 

856201 

2.04 

986354 

4 

21 

013646 

54 

I 

842685 

2 

n 

856078 

2-04 

986607 

4 

21 

013393 

53 

842815 

2 

i-j 

855956 
855833 

2. 04 

986860 

4 

21 

01 3 1 40 

52 

9 

842946 

2 

17 

2-04 

987112 

4 

21 

012888 

5i 

10 

843076 

2 

17 

8557 1 1 

2.o5 

987365 

4 

21 

012635 

5o  1 

11 

9-843206 

2 

16 

9-855588 

2-o5 

9-987618 

4 

21 

10-012382 

49 

12 

843336 

2 

16 

855465 

2-05 

987871 

4 

21 

012129 

48 

i3 

843466 

2 

16 

855342 

2-o5 

988123 

4 

21 

011877 

47 

14 

843595 

2 

16 

855219 

2-05 

988376 

4 

21 

011624 

46 

i5 

843725 

2 

16 

855096 

2-05 

988629 

4 

21 

011371 

45 

i6 

843855 

2 

16 

854973 

2-05 

988882 

4 

21 

011118 

44 

13 

843984 

2 

16 

854850 

2-o5 

989134 

4 

21 

010S66 

43 

8441 14 

2 

i5 

854727 

2-06 

989387 

4 

21 

oio6i3 

42 

19 

844243 

2 

i5 

854603 

2-06 

989640 

4 

21 

oio36o 

41 

20 

844372 

2 

i5 

854480 

2 -06 

989893 

4 

21 

010107 

40 

21 

9 -844502 

2 

i5 

9-854356 

2.06 

9-990145 

4 

21 

10-009855 

li 

S2 

844631 

2 

i5 

854233 

2 -06 

99o3gS 

4 

21 

009602 

23 

844760 

2 

i5 

854109 

2 -06 

99o6di 

4 

21 

009349 

u 

24 

844889 

2 

i5 

8539S6 

2 -06 

990903 

4 

21 

008844 

25 

8430 1  a 

2 

i5 

853562 

2-o6 

99 1 1 36 

4 

21 

35 

26 

843147 

2 

i5 

853738 

2-06 

991409 

4 

21 

008591 
008338 

34 

11 

845276 

2 

14 

853614 

2-07 

991662 

4 

21 

33 

845405 

2 

14 

853490 

2-07 

991914 

4 

21 

008086 

32 

29 

845533 

2 

14 

853366 

2-07 

992167 

4 

21 

007833 

3i 

3o 

845662 

2 

14 

853242 

2-07 

992420 

4 

21 

007580 

3o 

3i 

9-845790 

2 

14 

9-853ii8 

2-07 

9-992672 

4 

21 

10-007328 

lt\ 

32 

845919 

2 

14 

852994 

2-07 

992925 

4 

21 

007075 

33 

846047 

2 

14 

852^69 

2-07 

993178 

4 

21 

006822 

27 

34 

846175 

2 

14 

852743 

2-07 

993430 

4 

21 

006570 

26 

35 

846304 

2 

14 

852620 

2-07 

993683 

4 

21 

oo63 1 7 

25 

36 

846432 

2 

i3 

852496 

2-o8 

993936 

4 

21 

006064 

24 

ll 

846560 

2 

i3 

852371 

2-08 

994189 

4 

21 

0058 1 1 

23 

846688 

2 

i3 

852247 

2-08 

994441 

4 

21 

005559 

22 

39 

846816 

2 

i3 

852122 

2-08 

994694 

4 

21 

oo53o6 

21  1 

4o 

846944 

2 

i3 

851997 

2-08 

994947 

4 

21 

oo5o53 

20  : 

41 

9-847071 

2 

i3 

9.851872 

2-o8 

9-995199 

4 

21 

10-004801 

>9 

42 

847199 

2 

i3 

851747 

2-08 

995432 

4 

21 

004548 

18 

43 

847327 

2 

i3 

85i622 

2-08 

995703 

4 

21 

004295 

'7 

44 

847454 

2 

12 

85 1 497 

2-09 

995957 

4 

21 

004043 

16  i 

45 

847582 

2 

12 

85i372 

2-09 

996210 

4 

21 

003790 

i5  ' 

46 

847709 

2 

12 

851246 

2-09 

996463 

4 

21 

003537 

U  , 

47 

847836 

2 

12 

85II2I 

2-09 

996715 

.4 

21 

0032S5 

i3  j 

48 

847964 

2 

12 

850996 

209 

996968 

4 

21 

oo3o32 

12  1 

49 

848091 

2 

12 

850870 

2-09 

997221 

4 

21 

002779 

11 

5o 

848218 

2 

12 

850745 

2-09 

997473 

4 

21 

002327 

10 

5i 

9-848345 

2 

12 

9.850619 

2-09 

9.997726 

4 

21 

10-002274 

l\ 

52 

848472 

2 

II 

850493 

210 

997979 

4 

21 

002021 

53 

848599 

2 

11 

85o368 

2-10 

998231 

4 

21 

001769 

7  1 

54 

848726 
848852 

2 

II 

850242 

2-10 

998484  . 

4 

21 

ooi5i6 

6 

55 

2 

11 

85om6 

2-10 

998737 

4 

21 

001263 

5 

56 

848979 

2 

II 

849990 

2-10 

998989 

4 

21 

OOIOI 1 

4 

57 

849106 

2 

II 

849^64 

2-10 

999242 

4 

21 

000708 

3 

58 

849232 

2 

II 

849738 

2-10 

999493 

4 

21 

ooo5o5 

2 

59 

849359 

2 

II 

849611 

2-10 

999748 

4 

21 

000253 

I 

60 

849483 

2-11 

849485 

2-10 

10-000000 

4-21 

10-000000 

°  1 

■  Cosine 

D. 

Sine 

D. 

( 'otans:. 

D. 

Tang. 

M.  1 

(dl: 

\    n  r.'  n  n 

.T.<iit  0 

0 

A 

~t        r^ 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 

This  book  is  DUE  on  the  last  date  stamped  below. 


m  ^^ 

JUN  1  6  1977 

JUH  3  0  1977 

REC'D  ^     ^ 

RECEIVED 

OCT  29    ,, 

:^-.'6    1385 


"Vx;    Ktcjced 


Oi.s^> 


Form  L9-39, 050-8/65  (F623488)4939 


UNIVERSITY  of  CAJ.IFORNIA 

AT       \ 
LOS  ANGELES 
LIBRARY 


I 


LOCKED  oust 


